Carath\'{e}odory interpolation on the non-commutative polydisk
\i
The Carath\'{e}odory problem in the $N$-variable non-commutative Herglotz--Agler class and the Carath\'{e}odory--Fej\'{e}r problem in the $N$-variable non-commutative Schur--Agler class are posed. It is shown that the Carath\'{e}o\-dory (resp., Carath\'{e}odory--Fej\'{e}r) problem has a solution if and only if the non-commutative polynomial with given operator coefficients (the data of the problem indexed by an admissible set $\Lambda$) takes operator values with positive semidefinite real part (resp., contractive operator values) on $N$-tuples of $\Lambda$-jointly nilpotent contractive $n\times n$ matrices, for all $n\in\mathbb{N}$.
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\title[Carath\'{e}odory interpolation on the non-commutative polydisk]{Carath\'{e}odory interpolation on the non-commutative polydisk}
\author[Dmitry S. Kalyuzhny\u{\i}-Verbovetzki\u{\i}]{Dmitry S. Kalyuzhny\u{\i}-Verbovetzki\u{\i}}
\address{
Department of Mathematics\\
Ben-Gurion University of the Negev\\
P.O. Box 653, Beer-Sheva 84105\\
Israel}
\email{dmitryk@math.bgu.ac.il}
\thanks{The author was supported by the Center for Advanced Studies in Mathematics, Ben-Gurion University of the Negev.}
\subjclass[2000]{Primary 47A57, 47A13; Secondary  46L89, 47A20.}
\keywords{Interpolation, Carath\'{e}odory problem, Carath\'{e}odory--Fej\'{e}r problem, non-commutative, formal power series, dilation, completely positive map, Arveson extension theorem, jointly nilpotent operators.}

\begin{document}
\begin{abstract}
The Carath\'{e}odory problem in the $N$-variable non-commutative Herglotz--Agler class and the Carath\'{e}odory--Fej\'{e}r problem in the $N$-variable non-commutative Schur--Agler class are posed. It is shown that the Carath\'{e}o\-dory (resp., Carath\'{e}odory--Fej\'{e}r) problem has a solution if and only if the non-commutative polynomial with given operator coefficients (the data of the problem indexed by an admissible set $\Lambda$) takes operator values with positive semidefinite real part (resp., contractive operator values) on $N$-tuples of $\Lambda$-jointly nilpotent contractive $n\times n$ matrices, for all $n\in\mathbb{N}$. 
\end{abstract}
\maketitle

\section{Introduction}\label{sec:intro}
The classical Carath\'{e}odory interpolation problem is the following: given a sequence of complex numbers $c_0>0,\ c_1,\ldots,c_m$, find a holomorphic function $f(z)=f_0+f_1z+f_2z^2+\cdots$ on the open unit disk $\mathbb{D}$ whose values in $\mathbb{D}$ have positive real part (i.e., $f$ belongs to the \emph{Herglotz}, or \emph{Carath\'{e}odory, class} $\mathcal{H}_1$, where the subscript $1$ stands for the one-variable case) such that 
$$f_0=\frac{c_0}{2},\quad f_1=c_1, \ldots,  f_m=c_m.$$
This problem has been posed by Constantin Carath\'{e}odory in \cite{Car1,Car2} where the criteria of its solvability and of the uniqueness of its solution were presented. Toeplitz has noticed in \cite{Toep} that the original solvability criterion from \cite{Car1}, which was formulated in terms of convex bodies, admits the following formulation in terms of the coefficients $c_k,\ k=0,\ldots,m$:
the Carath\'{e}odory problem for these data has a solution if and only if the $(m+1)\times (m+1)$ matrix
	\begin{equation}\label{t}
	T_c=\left[
\begin{array}{lllll}
	c_0       & c_1^*    & \ldots & c_{m-1}^* & c_m^*\\
	c_1     & \ddots    & \ddots    & \ldots  & c_{m-1}^*\\
	\vdots    & \ddots & \ddots & \ddots  & \vdots\\
		c_{m-1} & \ldots & \ddots & \ddots     & c_1^*\\
	c_m     & c_{m-1} & \ldots & c_1 & c_0  
\end{array}\right] 
\end{equation}
 is positive semidefinite (here $c_k^*=\overline{c_k}$). From the integral representation by Riesz \cite{Ri} and Herglotz \cite{He}
 \begin{equation}\label{rh}
 f(z)=\frac{1}{2}\int_\mathbb{T}\frac{1+\bar{\lambda}z}{1-\bar{\lambda}z}\,d\mu(\lambda)+i\Im f(0),\quad z\in\mathbb{D},
 \end{equation}
 which characterizes functions from $\mathcal{H}_1$ (here $\mu$ is a positive Borel measure on the unit circle $\mathbb{T}$; in the case where $f(0)=\frac{1}{2}$ the second term in the right-hand side of \eqref{rh} is dropped out and $\mu$ has full variation $|\mu|=1$) one obtains a representation for the Taylor coefficients of $f\in\mathcal{H}_1$:
 $$f_0=\frac{|\mu|}{2}+i\Im f(0),\quad f_k=\int_\mathbb{T}\bar{\lambda}^k\,d\mu(\lambda),\quad k=1,2,\ldots .$$
Thus, the Carath\'{e}odory problem has a solution if and only if there exists a positive Borel measure $\mu$ on $\mathbb{T}$ such that
\begin{equation}\label{mom}
c_k=\int_\mathbb{T}\bar{\lambda}^k\,d\mu(\lambda),\quad k=0,\ldots, m,
\end{equation}
 i.e., $\mu$ solves the \emph{trigonometric moment problem} for the data $c_k,\ k=0,\ldots,m$.
 
 In the operator case the data of the Carath\'{e}odory problem are bounded linear operators $c_0\geq 0,\ c_1,\ldots,c_m$ on a separable Hilbert space\footnote{In this paper we will consider separable Hilbert spaces only, and omit ``separable" for brevity.} $\mathcal{Y}$, and the class $\mathcal{H}_1$ is replaced by the class $\mathcal{H}_1(\mathcal{Y})$ of holomorphic functions on $\mathbb{D}$ whose values are bounded linear operators\footnote{For Hilbert spaces $\mathcal{Y}$ and $\mathcal{H}$, we shall use the notation $\mathcal{L(Y,H)}$ (resp., $\mathcal{L(Y)}$) for the Banach space of bounded linear operators from $\mathcal{Y}$ to $\mathcal{H}$ (resp., from $\mathcal{Y}$ to itself).} on $\mathcal{Y}$ with positive semidefinite real part. Then the Carath\'{e}odory--Toeplitz criterion, representation \eqref{rh} for $f\in\mathcal{H}_1(\mathcal{Y})$, and trigonometric moment representation \eqref{mom} hold true with the operator block matrix $T_c$ in \eqref{t}, a positive Borel $\mathcal{L(Y)}$-valued measure $\mu$, and the convergence of integrals in \eqref{rh} and \eqref{mom} in the strong operator topology. Riesz--Herglotz representation \eqref{rh} for the case where $f(0)=\frac{I_\mathcal{Y}}{2}$, and thus moment representation \eqref{mom} for the case where $c_0=I_\mathcal{Y}$ admit the following operator form:
\begin{eqnarray}
f(z) &=& \frac{1}{2}V^*(I_\mathcal{H}+zG)(I_\mathcal{H}-zG)^{-1}V,\quad z\in\mathbb{D}, \label{rh-o}\\
c_k  &=& V^*G^kV,\qquad k=0,\ldots,m,\label{mom-o}
\end{eqnarray}
 where $G$ is a unitary operator on some auxiliary Hilbert space $\mathcal{H}$, and $V\in\mathcal{L(Y,H)}$ is an isometry. These results are due to Neumark \cite{Ne}.
 
A similar problem was considered first by Carath\'{e}odory and Fej\'{e}r in \cite{CF} for the \emph{Schur class} $\mathcal{S}_1$ of holomorphic contractive functions on $\mathbb{D}$ in the place of the Herglotz class $\mathcal{H}_1$:  given a sequence of complex numbers $s_0,\ldots,s_m$, find a holomorphic function $F(z)=F_0+F_1z+F_2z^2+\cdots$ from the class $\mathcal{S}_1$ such that 
$$F_0=s_0,\ldots,F_m=s_m.$$
Schur has proved in \cite{Sch} that the Carath\'{e}odory--Fej\'{e}r problem has a solution if and only if the matrix
	\begin{equation}\label{st}
	T_s=\left[
\begin{array}{llll}
	s_0       & 0    & \ldots & 0\\
	s_1     & \ddots   & \ddots    & \vdots \\
	\vdots    & \ddots & \ddots & 0\\
	s_m     & \ldots  & s_1 & s_0  
\end{array}\right] 
\end{equation}
is contractive\footnote{Matrix norm considered in this paper is operator $(2,2)$-norm, i.e., the maximal singular value of a matrix.}, i.e., $\| T_s\|\leq 1$. In the operator case the data of the Carath\'{e}odory--Fej\'{e}r problem are operators $s_0,\ldots,s_m\in\mathcal{L(U,Y)}$, with  Hilbert spaces $\mathcal{U}$ and $\mathcal{Y}$, the class $\mathcal{S}_1$ is replaced by the class $\mathcal{S}_1(\mathcal{U,Y})$ of holomorphic functions on $\mathbb{D}$ whose values are contractive operators from $\mathcal{L(U,Y)}$ and the Schur criterion is formulated in the same way as in the scalar case, with the operator block matrix $T_s$.

Let us note that a common operatorial view at the Carath\'{e}odory problem, Carath\'{e}odory--Fej\'{e}r problem, and their relative Nevanlinna--Pick problem, and a certain operator dilation scheme which unifies these problems were first presented in the fundamental paper of Sarason \cite{Sar}. These ideas in an abstract form have been expressed in the commutant lifting theorem of Sz.-Nagy and Foia\c{s} (see \cite{SzNF}) which is used now as one of the approaches to various interpolation problems. For further details on the classical and operator versions of the Carath\'{e}odory problem and other interpolation problems, see \cite{Akh, KN, FF,Co, Sakh}.

There exist various generalizations of the Carath\'{e}odory problem and other interpolation problems to the case of several complex variables, depending on the type of a classical domain in $\mathbb{C}^N$ serving as a counterpart of $\mathbb{D}$ and on the class of interpolating functions. Due to a version of the Riesz--Herglotz formula \eqref{rh} for
the unit polydisk $\mathbb{D}^N$ obtained by Kor\'{a}nyi and Puk\'{a}nszky \cite{KP}, one can characterize the coefficients of a function from the multivariable \emph{Herglotz class} $\mathcal{H}_N(\mathcal{Y})$ (the class of holomorphic functions on $\mathbb{D}^N$ taking operator values from $\mathcal{L(Y)}$ with positive semidefinite real part) in terms of a $\mathcal{L(Y)}$-valued positive Borel measure $\mu$ whose Fourier coefficients with multi-indices outside $\mathbb{Z}^N_+$ and $\mathbb{Z}^N_-$, the positive and the negative discrete octants, are zero. However, an appropriate multivariable analogue of \eqref{rh-o} (and thus, of \eqref{mom-o}) can be obtained either for the case $N=2$ or for the subclass $\mathcal{HA}_N(\mathcal{Y})\subset\mathcal{H}_N(\mathcal{Y})$ which is proper for $N>2$. The latter subclass, which is called the  \emph{Herglotz--Agler class}, has been introduced by Agler in \cite{Ag1}, where the analogue of \eqref{rh-o} has been obtained. This class $\mathcal{HA}_N(\mathcal{Y})$ consists of holomorphic $\mathcal{L(Y)}$-valued functions on $\mathbb{D}^N$ whose values on any $N$-tuple of commuting strict contractions on a common Hilbert space (in the sense of hereditary functional calculus introduced in \cite{Ag1}) have positive semidefinite real part. Some partial results on the Carath\'{e}odory--Fej\'{e}r problem in the class $\mathcal{S}_N$ (the \emph{Schur class} of contractive holomorphic functions on $\mathbb{D}^N$) have been obtained in \cite{Pf,DKh}. The Carath\'{e}odory and Carath\'{e}odory--Fej\'{e}r problems in the Herglotz--Agler class $\mathcal{HA}_N(\mathcal{Y})$
and the \emph{Schur--Agler class $\mathcal{SA}_N(\mathcal{U,Y})$} (the class of holomorphic $\mathcal{L(U,Y)}$-valued functions on $\mathbb{D}^N$ which take contractive operator values on any $N$-tuple of commuting strict contractions, in the sense of Agler's hereditary functional calculus), respectively, were studied in \cite{EPP,BLTT,W}. Various versions of the Sarason theorem, the Sz.-Nagy--Foia\c{s} commutant lifting theorem and the Nevanlinna--Pick problem on $\mathbb{D}^N$, in the classes $\mathcal{H}_N(\mathcal{Y}), \mathcal{HA}_N(\mathcal{Y}), \mathcal{S}_N(\mathcal{U,Y})$ and $\mathcal{SA}_N(\mathcal{U,Y})$ were considered in \cite{CS,CW,AgM1,BT,BLTT,EPP}. The Kor\'{a}nyi--Puk\'{a}nszky version of the Riesz--Herglotz formula has been generalized in \cite{AD} to a wide class of domains in $\mathbb{C}^N$ which contains, in particular, all classical symmetric domains. Certain partial results on the Carath\'{e}odory--Fej\'{e}r problem for bounded full circular domains in $\mathbb{C}^N$ can be found in \cite{DKh}. A generalization of  
the Agler representation theorem from \cite{Ag1} to a class of so-called polynomially defined domains in $\mathbb{C}^N$ has been obtained in \cite{AT,BB1} where also the Nevanlinna--Pick problem in the Schur--Agler class of functions on such a domain was studied. The Nevanlinna--Pick and Carath\'{e}odory--Fej\'{e}r problems in the class of contractive multipliers on the reproducing kernel Hilbert space of holomorphic functions on the \emph{unit ball} $\mathbb{B}_N:=\{ z\in\mathbb{C}^N:\ \sum_{k=1}^N|z_k|^2<1\}$, with the reproducing kernel $k_N(z,z')=\frac{1}{1-\left\langle z,z'\right\rangle}$, or more generally, on the reproducing kernel Hilbert space of functions on a set $\Omega$, with the reproducing kernel whose reciprocal has exactly one positive square, were studied starting with the unpublished paper of Agler \cite{Ag2} by many authors (e.g., see \cite{Ag3,McC1,McC2,Q,AgM2,BTV,BB2}). Let us mention also the approach to interpolation problems on $\mathbb{B}_N$ via the commutant lifting theorem in the non-commutative setting of the Toeplitz algebra of operators acting on the Fock space by Popescu \cite{P1,P2} and subsequent use of symmetrization argument (see \cite{P3,AP,DP}). In this non-commutative setting the Carath\'{e}odory--Fej\'{e}r problem was studied in \cite{P3,P4,CoJ}. (A certain generalization of Popescu's non-commutative setting and a more general Nevanlinna--Pick interpolation problem appears in a recent paper \cite{MS}.) Let us remark that one can interprete the latter results in terms of functions on the \emph{non-commutative unit ball} $\mathcal{B}_N$ which is the collection of \emph{strict row contractions}, i.e., $N$-tuples of bounded linear operators $\mathbf{T}=(T_1,\ldots,T_N)$ on a common Hilbert space $\mathcal{E}$ such that $\sum_{k=1}^NT_kT_k^*<I_\mathcal{E}$.

In the present paper, we are working on another domain, the \emph{non-commutative unit polydisk} $\mathcal{D}^N$ which is the collection of $N$-tuples $\mathbf{T}=(T_1,\ldots,T_N)$ of strict contractions on a common Hilbert space $\mathcal{E}$, i.e, $\| T_k\|<1,\ k=1,\ldots,N$, or on the \emph{non-commutative matrix unit polydisk}, which is a subbdomain $\mathcal{D}^N_{\rm matr}\subset\mathcal{D}^N$ consisting of $N$-tuples of strict contractions on $\mathbb{C}^n$, for all $n\in\mathbb{N}$. The domain $\mathcal{D}^N$ is a special case of a bit more general non-commutative domain $\mathcal{D}_G$ considered in the recent paper of Ball, Groenewald and Malakorn \cite{BGM} where the \emph{non-commutative Schur--Agler class} $\mathcal{SA}_N^{\rm nc}(\mathcal{U,Y})$ was introduced and studied in the framework of structured non-commutative multidimensional conservative linear systems. The domain $\mathcal{D}^N_{\rm matr}$ appears in \cite{AK2}. We consider non-commutative formal power series which converge on $\mathcal{D}^N$. We introduce the \emph{non-commutative Herglotz--Agler class} $\mathcal{HA}_N^{\rm nc}(\mathcal{Y})$ of such series which take on $\mathcal{D}^N$ operator values with positive semidefinite real part, and study the Carath\'{e}odory problem in this class, as well as the Carath\'{e}odory--Fej\'{e}r problem in the class $\mathcal{SA}_N^{\rm nc}(\mathcal{U,Y})$.

To give an idea on our main results, criteria of solvability of these problems, let us first come back to the one-variable case. Let $S$ denote the standard shift $(m+1)\times(m+1)$ matrix:
\begin{equation}\label{s}
S=\left[\begin{array}{cccc}
0 &        &     & \\
1 & \ddots &     & \\
  & \ddots & \ddots & \\
  &        &     1  & 0 
\end{array}\right],
\end{equation}
i.e., $S_{ij}=1$ for $i-j=1$, and $S_{ij}=0$ otherwise. Then
$$T_c=I_{m+1}\otimes c_0+\sum_{k=1}^mS^k\otimes c_k+\sum_{k=1}^mS^{*k}\otimes c_k^*.$$
If one defines
$$p(z):=\frac{c_0}{2}+\sum_{k=1}^mc_kz^k,\qquad z\in\mathbb{C},$$
then the Carath\'{e}odory--Toeplitz criterion can be formulated as the positive semidefiniteness of the operator $2\Re p^{\rm l}(S)=p^{\rm l}(S)+p^{\rm l}(S)^*$, where
$$p^{\rm l}(S):=\frac{I_{m+1}\otimes c_0}{2}+\sum_{k=1}^mS^k\otimes c_k,$$
or equivalently, of the operator $2\Re p(S)$, where
$$p(S)=p^{\rm r}(S):=\frac{c_0\otimes I_{m+1}}{2}+\sum_{k=1}^mc_k\otimes S^k$$
(we shall usually omit the superscript ``r", however keep the superscript ``l" when we use the writing of a polynomial with powers on the left). By \cite[Section~2.5]{Arv2}, any contraction $T$ on a Hilbert space $\mathcal{E}$ which is \emph{nilpotent of rank at most $m+1$}, i.e., such that 
$$T^k=0,\qquad k=m+1,m+2,\ldots,$$
admits a \emph{dilation} of the form $S\otimes I_\mathcal{H}$, with some Hilbert space $\mathcal{H}$, i.e., there exists an isometry $V\in\mathcal{L(E},\mathbb{C}^{m+1}\otimes\mathcal{H})$ such that
$$T^k=V^*(S^k\otimes I_\mathcal{H})V,\qquad k=1,2,\ldots .$$
Since $S$ is a nilpotent matrix with rank of nilpotency $m+1$, we obtain the following criterion:
\emph{the Carath\'{e}odory problem with data $c_0\geq 0,c_1,\ldots,c_m\in\mathcal{L(Y)}$ has a solution if and only if $\Re p(T)\geq 0$ for every nilpotent operator $T$ with rank of nilpotency at most $m+1$.} 
Analogously, $$T_s=\sum_{k=0}^mS^k\otimes s_k.$$
If one defines
$$q(z):=\sum_{k=0}^ms_kz^k,\qquad z\in\mathbb{C},$$
then the Schur criterion can be formulated as the contractivity of the operator $q^{\rm l}(S)$ where
$$q^{\rm l}(S)=\sum_{k=0}^mS^k\otimes s_k,$$
or equivalently, of the operator $q(S)$, where
$$q(S)=\sum_{k=0}^ms_k\otimes S^k.$$
Thus, \emph{the Carath\'{e}odory--Fej\'{e}r problem with data $s_0,\ldots,s_m\in\mathcal{L(U,Y)}$ has a solution if and only if $\| q(T)\|\leq 1$ for every nilpotent operator $T$ with rank of nilpotency at most $m+1$.}

The main results of the paper are generalizations of these criteria to the multivariable non-commutative case, where the positivity or contractivity of a non-commutative polynomial is tested on $N$-tuples of \emph{jointly nilpotent} contractions $\mathbf{T}=(T_1,\ldots,T_N)$, i.e, $\| T_k\|\leq 1,\  k=1,\ldots,N,$ and $T_{i_1}\cdots T_{i_k}=0$ outside some finite set of strings $(i_1,\ldots,i_k),\ k\in\mathbb{N}$. To obtain these criteria, we first deduce the analogue of \eqref{rh-o} for non-commutative formal power series of the class $\mathcal{HA}_N^{\rm nc}(\mathcal{Y})$ from the realization formula obtained in \cite{BGM} for the class $\mathcal{SA}_N^{\rm nc}(\mathcal{Y})$. Thus, an anologue of \eqref{mom-o} is also obtained. A counterpart of unitary operator $G$ from \eqref{rh-o} and \eqref{mom-o} is $N$-tuple
$\mathbf{G}=(G_1,\ldots,G_N)$ of bounded linear operators on a common Hilbert space satisfying the following condition:
\begin{equation}\label{g}
\zeta\mathbf{G}:=\sum_{k=1}^N\zeta_kG_k\quad {\rm is\ unitary\ for\ every}\ \zeta\in\mathbb{T}^N.
\end{equation} 
We denote by $\mathcal{G}_N$ the class of such $N$-tuples of operators. Note that an $N$-tuple $\mathbf{G}$ from the class  $\mathcal{G}_N$ appears also in Agler's representation formula for functions from the (commutative) class $\mathcal{HA}_N(\mathcal{Y})$ in \cite{Ag1}, and in a realization formula for functions from the subclass $\mathcal{SA}_N^0(\mathcal{U,Y})\subset\mathcal{SA}_N(\mathcal{U,Y})$ which consists of functions vanishing at zero, in \cite{K2}. We deduce the criterion of solvability of the Carath\'{e}odory--Fej\'{e}r problem in the class $\mathcal{SA}_N^{\rm nc}(\mathcal{U,Y})$  from the one for the Carath\'{e}odory problem in the class $\mathcal{HA}_N^{\rm nc}(\mathcal{Y})$, which we obtain first.

The main tools in our work, besides the realization formula from \cite{BGM} mentioned above, are the following: the properties of operator $N$-tuples from the class $\mathcal{G}_N$ which have been established in \cite{K2} and some their new properies which we obtain in the present paper; the factorization result of McCullough \cite{McC3} for non-commutative hereditary polynomials; the Arveson extension theorem \cite{Arv1}; the Stinespring representation theorem \cite{St} for completely positive maps of $C^*$-algebras; the Sz.-Nagy and Foia\c{s} theorem on the existence of a unitary dilation of an $N$-tuple of (not necessarily commuting) contractions \cite{SzNF}; the Amitsur--Levitzki theorem  on the non-existence of non-commutative polynomial relations valid for infinitely many matrix rings $\mathbb{C}^{n_j\times n_j},\ j=1,2,\ldots$ (see, e.g., \cite[pp. 22--23]{Ro}).

The structure of the paper is the following. In Section~\ref{sec:tuples} we study certain classes of operator $N$-tuples. In particular, we establish duality properties of the classes $\mathcal{G}_N$ and $\mathcal{U}^N$. The latter is the class of $N$-tuples $\mathbf{U}=(U_1,\ldots,U_N)$ of unitary operators on a common Hilbert space, which serves as another generalization (in addition to $\mathcal{G}_N$) of the class of single unitaries. This duality is observed also in Lemma~\ref{lem:main} which is proved in Section~\ref{sec:prob}. In Section~\ref{sec:ha} we introduce and characterize the non-commutative Herglotz--Agler class $\mathcal{HA}_N^{\rm nc}(\mathcal{Y})$. In Section~\ref{sec:prob} we formulate the Carath\'{e}odory problem in the class  $\mathcal{HA}_N^{\rm nc}(\mathcal{Y})$ and prove the necessary and sufficient conditions for its solvability. In Section~\ref{sec:prob-cf} we formulate the Carath\'{e}odory--Fej\'{e}r problem in the class  $\mathcal{SA}_N^{\rm nc}(\mathcal{U,Y})$ and obtain a criterion of its solvability.
 
\section{Some classes of operator $N$-tuples}\label{sec:tuples}
Let us define the classes of $N$-tuples of operators which are considered in this paper. In addition to the classes $\mathcal{D}^N$, $\mathcal{D}^N_{\rm matr}$, $\mathcal{G}_N$ and $\mathcal{U}^N$ already mentioned in Section~\ref{sec:intro}, let us define the class $\mathcal{C}^N$ which consists of $N$-tuples $\mathbf{C}=(C_1,\ldots,C_N)$ of contractions on a common Hilbert space, i.e., $\| C_k\|\leq 1,\ k=1,\ldots,N$. The class $\mathcal{G}_N$ is characterized by the following proposition which is a consequence of \cite[Proposition 2.4]{K2}.
\begin{prop}\label{prop:gn}
Let $\mathbf{G}=(G_1,\ldots,G_N)\in\mathcal{L(H)}^N$, with a Hilbert space $\mathcal{H}$. The following statements are equivalent:
\allowdisplaybreaks
\begin{description}
\item[(i)] $\mathbf{G}\in\mathcal{G}_N$;
	\item[(ii)] $\mathbf{G}$ satisfies the conditions
	\begin{eqnarray}
	\sum_{k=1}^NG_k^*G_k &=& I_\mathcal{H},\label{i1}\\
	G_k^*G_j &=& 0,\qquad k\neq j,\label{i2}\\
	\sum_{k=1}^NG_kG_k^* &=& I_\mathcal{H},\label{i3}\\
	G_kG_j^* &=& 0,\qquad k\neq j;\label{i4}
		\end{eqnarray} 
		\item[(iii)] the operator $G^0:=\sum_{k=1}^NG_k$ is unitary, and there exists a resolution of identity $I_\mathcal{H}=\sum_{k=1}^NP_k^-$, where $(P_k^-)^2=P_k^-=(P_k^-)^*,\ k=1,\ldots,N$, and $P_k^-P_j^-=0$ for $k\neq j$, such that
		$$G_k=G^0P_k^-,\qquad k=1,\ldots,N;$$
		 	\item[(iv)] the operator $G^0:=\sum_{k=1}^NG_k$ is unitary, and there exists a resolution of identity $I_\mathcal{H}=\sum_{k=1}^NP_k^+$, where $(P_k^+)^2=P_k^+=(P_k^+)^*,\ k=1,\ldots,N$, and $P_k^+P_j^+=0$ for $k\neq j$, such that
		$$G_k=P_k^+G^0,\qquad k=1,\ldots,N;$$
				\item[(v)] for every $\mathbf{U}=(U_1,\ldots,U_N)\in\mathcal{U}^N\cap\mathcal{L}(\mathcal{K})^N$, with a Hilbert space $\mathcal{K}$, the operator
		\begin{equation}\label{l-pencil}
		\mathbf{U\otimes G}:=\sum_{k=1}^NU_k\otimes G_k\in\mathcal{L}(\mathcal{K}\otimes\mathcal{H})
		\end{equation}
		is unitary;
		\item[(vi)] for every $\mathbf{U}=(U_1,\ldots,U_N)\in\mathcal{U}^N\cap\mathcal{L}(\mathcal{K})^N$, with a Hilbert space $\mathcal{K}$, the operator
		\begin{equation}\label{r-pencil}
		\mathbf{G\otimes U}:=\sum_{k=1}^NG_k\otimes U_k\in\mathcal{L}(\mathcal{H\otimes K})
		\end{equation}
		is unitary.
		\end{description}
\end{prop}
\begin{cor}\label{cor:gn}
If $\mathbf{G}\in\mathcal{G}_N\cap\mathcal{L(H)}^N$ then: 
\begin{description}
\item[(a)] for every $\mathbf{C}\in\mathcal{C}^N\cap\mathcal{L}(\mathcal{E})^N$, with a Hilbert space $\mathcal{E}$, the operators $\mathbf{G\otimes C}$ and $\mathbf{C\otimes G}$ are contractions; 
\item[(b)]  for every $\mathbf{C}\in\mathcal{D}^N\cap\mathcal{L}(\mathcal{E})^N$, with a Hilbert space $\mathcal{E}$, the operators $\mathbf{G\otimes C}$ and $\mathbf{C\otimes G}$ are strict contractions.
\end{description}
\end{cor}
\begin{proof}
(a) Any $\mathbf{C}\in\mathcal{C}^N\cap\mathcal{L}(\mathcal{E})^N$ has a \emph{unitary dilation} \cite{SzNF}, i.e., there exists an $N$-tuple  $\mathbf{U}\in\mathcal{U}^N\cap\mathcal{L(K)}^N$, with some Hilbert space $\mathcal{K}\supset\mathcal{E}$, such that
$$C_{i_1}\cdots C_{i_l}=P_\mathcal{E}U_{i_1}\cdots U_{i_l}\big|_\mathcal{E}, \qquad l\in\mathbb{N},\ i_1,\ldots,i_l\in\{ 1,\ldots,N\},
$$
where $P_\mathcal{E}$ denotes the orthogonal projection onto the subspace $\mathcal{E}$ in $\mathcal{K}$. Therefore, $$\mathbf{G\otimes C}=\sum_{k=1}^NG_k\otimes P_\mathcal{E}U_k\big|_\mathcal{E}=(I_\mathcal{H}\otimes P_\mathcal{E})(\mathbf{G\otimes U})\big|_{\mathcal{H}\otimes\mathcal{E}},$$ and since by Proposition~\ref{prop:gn} $\mathbf{G\otimes U}$ is unitary, $\mathbf{G\otimes C}$ is a contraction. Analogously, $\mathbf{C\otimes G}$ is a contraction.

(b) If $\mathbf{C}\in\mathcal{D}^N\cap\mathcal{L}(\mathcal{E})^N$ is non-zero (otherwise the statement is trivial) then $\widetilde{\mathbf{C}}:=(\max_{1\leq k\leq N}\| C_k\| )^{-1}\mathbf{C}\in\mathcal{C}^N\cap\mathcal{L}(\mathcal{E})^N$. By part (a) of this Proposition, $\mathbf{G}\otimes \widetilde{\mathbf{C}}$ and $\widetilde{\mathbf{C}}\otimes \mathbf{G}$ are contractions. Therefore, 
 $\mathbf{G\otimes C}$ and $\mathbf{C\otimes G}$ are strict contractions with norm at most $\max_{1\leq k\leq N}\| C_k\|$.
\end{proof}
The following proposition is dual to Proposition~\ref{prop:gn}.
\begin{prop}\label{prop:un} 
Let $\mathbf{U}=(U_1,\ldots,U_N)\in\mathcal{L(K)}^N$, with a Hilbert space $\mathcal{K}$. The following statements are equivalent:
\begin{description}
	\item[(i)] $\mathbf{U}\in\mathcal{U}^N$;
	\item[(ii)] for every $\mathbf{G}\in\mathcal{G}_N\cap\mathcal{L}(\mathcal{H})^N$, with a Hilbert space $\mathcal{H}$, the operator $\mathbf{U\otimes G}\in\mathcal{L}(\mathcal{K}\otimes\mathcal{H})^N$ is unitary;
	\item[(iii)] for every $\mathbf{G}\in\mathcal{G}_N\cap\mathcal{L}(\mathcal{H})^N$, with a Hilbert space $\mathcal{H}$, the operator $\mathbf{G\otimes U}\in\mathcal{L}(\mathcal{H}\otimes\mathcal{K})^N$ is unitary.
\end{description}
\end{prop}
\begin{proof}
If $\mathbf{U}\in\mathcal{U}^N$ and $\mathbf{G}\in\mathcal{G}_N$ then implications (i)$\Rightarrow$(ii) and (i)$\Rightarrow$(iii) follow from Proposition~\ref{prop:gn}.
For the proof of (ii)$\Rightarrow$(i) and (iii)$\Rightarrow$(i), one can choose $$\mathbf{G}^{(k)}:=(0,\ldots,0,1,0,\ldots, 0)\in\mathbb{C}^N\cong\mathcal{L}(\mathbb{C})^N,$$ where $1$ is on the $k$'th position and $0$ is on the other positions. It is clear that $ \mathbf{G}^{(k)}\in\mathcal{G}_N$. Since for every $k\in\{ 1,\ldots,N\}$ the operator $U_k=\mathbf{U\otimes G}^{(k)}=\mathbf{G}^{(k)}\otimes\mathbf{U}$ is unitary, $\mathbf{U}\in\mathcal{U}^N$. 
\end{proof}
For $N$-tuples of operators $\mathbf{X}=(X_1,\ldots,X_N)\in\mathcal{L(X)}^N$ and $\mathbf{Y}=(Y_1,\ldots,Y_N)\in\mathcal{L(Y)}^N$ on Hilbert spaces $\mathcal{X}$ and $\mathcal{Y}$, respectively, define their \emph{Schur tensor product} as the $N$-tuple of operators 
\begin{equation}\label{schur}
\mathbf{X}\stackrel{\circ}{\otimes}\mathbf{Y}:=(X_1\otimes Y_1,\ldots,X_N\otimes Y_N)\in\mathcal{L(X\otimes Y)}^N.
\end{equation}
\begin{prop}\label{prop:schur}
For any $\mathbf{G}\in\mathcal{G}^N$ and $\mathbf{U}\in\mathcal{U}^N$ both $\mathbf{G}\stackrel{\circ}{\otimes}\mathbf{U}$ and $\mathbf{U}\stackrel{\circ}{\otimes}\mathbf{G}$ belong to the class $\mathcal{G}^N$.
\end{prop}
\begin{proof}
For an arbitrary $\tilde{\mathbf{U}}\in\mathcal{U}^N$ the operator
$$(\mathbf{G}\stackrel{\circ}{\otimes}\mathbf{U})\otimes\tilde{\mathbf{U}}=\sum_{k=1}^NG_k\otimes U_k\otimes \tilde{U}_k=\mathbf{G}\otimes (\mathbf{U}\stackrel{\circ}{\otimes}\tilde{\mathbf{U}})$$
is unitary, by Proposition~\ref{prop:gn} and due to the fact that $U_k\otimes \tilde{U}_k$ are unitary operators for all $k=1,\ldots,N$, i.e., $\mathbf{U}\stackrel{\circ}{\otimes}\tilde{\mathbf{U}}\in\mathcal{U}^N$. Thus, again by Proposition~\ref{prop:gn}, $\mathbf{G}\stackrel{\circ}{\otimes}\mathbf{U}\in\mathcal{G}^N$. Analogously, $\mathbf{U}\stackrel{\circ}{\otimes}\mathbf{G}\in\mathcal{G}^N$. 
\end{proof}
Let us note that for the classes introduced above the following inclusions hold: 
$$\begin{array}{ccccccc}
                         &           &               &         & \mathcal{G}_N &         &       \\        
                         &           &               &         &       \cap    &         &       \\  
\mathcal{D}^N_{\rm matr} & \subset   & \mathcal{D}^N & \subset & \mathcal{C}^N & \supset & \mathcal{U}^N.
\end{array}$$
A couple of additional classes of operator $N$-tuples will be considered in Section~\ref{sec:prob}.  


\section{The non-commutative Herglotz--Agler class}\label{sec:ha}
 Let us give some necessary definitions in our non-commutative setting. The \emph{free semigroup $\mathcal{F}_N$ with the generators $g_1,\ldots,g_N$ (the letters)} and the neutral element $\emptyset$ (\emph{empty word}) has a product defined as follow: if two its elements (\emph{words}) are given by $w=g_{i_1}\cdots g_{i_m}$ and  $w'=g_{j_1}\cdots g_{j_{m'}}$ then their product is $ww'=g_{i_1}\cdots g_{i_m}g_{j_1}\cdots g_{j_{m'}},$
and  $w\emptyset=\emptyset w=w$. The \emph{length of the word} $w=g_{i_1}\cdots g_{i_m}$ is $|w|=m$, and $|\emptyset|=0$. 
The non-commutative algebra $\mathcal{L(Y)}\left\langle \left\langle z_1,\ldots,z_N\right\rangle\right\rangle$ consists 
of \emph{formal power series} $f$ with coefficients $f_w\in\mathcal{L(Y)},\ w\in\mathcal{F}_N$, for a Hilbert space $\mathcal{Y}$, of the form
$$f(z)=\sum_{w\in\mathcal{F}_N}f_wz^w,$$
where for the indeterminates $z=(z_1,\ldots,z_N)$ and  words $w=g_{i_1}\cdots g_{i_m}$ one sets $z^w=z_{i_1}\cdots z_{i_m},\ z^\emptyset =1$. We assume that indeterminates $z_k$ formally commute with coefficients $f_w$. A formal power series $f$ is invertible in this algebra if and only if $f_\emptyset$ is inverible. Indeed, if $f(z)\phi(z)=\phi(z)f(z)=I_\mathcal{Y}$ then $f_\emptyset\phi_\emptyset=\phi_\emptyset f_\emptyset=I_\mathcal{Y}$, i.e., $\phi_\emptyset=f_\emptyset^{-1}$. Conversely, if $f_\emptyset$ is inverible then the series
 $$
\phi(z)=\sum_{k=0}^\infty \left(I_\mathcal{Y}-
f_\emptyset^{-1}f(z)\right)^kf_\emptyset^{-1}.
$$
is the inverse of $f$.
This formal power series is well defined since the
expansion of $\left(I_\mathcal{Y}-f_\emptyset^{-1}f\right)^k$ contains words of length
at least $k$, and thus the expressions for coefficients  $\phi_w$ are finite
sums. The subalgebra $\mathcal{L(Y)}\left\langle z_1,\ldots,z_N\right\rangle$ of the algebra $\mathcal{L(Y)}\left\langle \left\langle z_1,\ldots,z_N\right\rangle\right\rangle$ consists 
of \emph{non-commutative polynomials} $p$  of the form
$$p(z)=\sum_{w\in\Lambda}p_wz^w,$$
where $\Lambda\subset\mathcal{F}_N$ is a finite set. We will consider also the space $\mathcal{L(U,Y)}\left\langle \left\langle z_1,\ldots,z_N\right\rangle\right\rangle$ of formal power series with coefficients in $\mathcal{L(U,Y)}$ and the space $\mathcal{L(U,Y)} \left\langle z_1,\ldots,z_N\right\rangle$ of non-commutative polynomials with coefficients in $\mathcal{L(U,Y)}$.

Let us introduce now the \emph{non-commutative Herglotz--Agler class} $\mathcal{HA}_N^{\rm nc}(\mathcal{Y})$ of formal power series $f\in\mathcal{L(Y)}\left\langle \left\langle z_1,\ldots,z_N\right\rangle\right\rangle$ such that 
the series
$$f(\mathbf{C}):=\sum_{w\in\mathcal{F}_N}f_w\otimes\mathbf{C}^w$$
converges in the operator norm and
$\Re f(\mathbf{C})\geq 0$ for every $\mathbf{C}\in\mathcal{D}^N\cap\mathcal{L(E)}^N$, with a Hilbert space $\mathcal{E}$. Here for $w=g_{i_1}\cdots g_{i_m}\in\mathcal{F}_N$ we set $\mathbf{C}^w:=C_{i_1}\cdots C_{i_m}$, and $\mathbf{C}^\emptyset =I_\mathcal{E}$. The subclass $\mathcal{HA}_N^{{\rm nc},I}(\mathcal{Y})\subset\mathcal{HA}_N^{\rm nc}(\mathcal{Y})$ consists of formal power series $f$ such that $f_\emptyset =I_\mathcal{Y}$.
\begin{thm}\label{thm:ha-nc}
A formal power series $f\in\mathcal{L(Y)}\left\langle \left\langle z_1,\ldots,z_N\right\rangle\right\rangle$ belongs to the class $\mathcal{HA}^{nc,I}_N(\mathcal{Y})$ if and only if there exist a Hilbert space $\mathcal{H}$, an $N$-tuple $\mathbf{G}\in\mathcal{G}_N\cap\mathcal{L(H)}^N$, and an isometry $V\in\mathcal{L(Y,H)}$ such that
\begin{equation}\label{ha-nc}
f(z)=V^*(I_\mathcal{H}+z\mathbf{G})(I_\mathcal{H}-z\mathbf{G})^{-1}V,
\end{equation}
where $z\mathbf{G}:=\sum_{k=1}^Nz_kG_k$ and thus
\begin{equation}\label{res}
(I_\mathcal{H}-z\mathbf{G})^{-1}=\sum_{j=0}^\infty\left(\sum_{k=1}^Nz_kG_k\right)^j=\sum_{w\in\mathcal{F}_N}\mathbf{G}^wz^w.
\end{equation}
\end{thm}
\begin{proof}
If \eqref{ha-nc} holds then $f\in\mathcal{HA}^{nc,I}_N(\mathcal{Y})$. Indeed, for any $\mathbf{C}\in\mathcal{D}^N\cap\mathcal{L(E)}^N$, with a Hilbert space $\mathcal{E}$, by Corollary~\ref{cor:gn} the operator  $\mathbf{G\otimes C}$ is a strict contraction. Then the series in \eqref{res} evaluated at $\mathbf{C}$ converges in the operator norm, thus the operator $(I_\mathcal{H\otimes E}+\mathbf{G\otimes C})(I_\mathcal{H\otimes E}-\mathbf{G\otimes C})^{-1}$ is well defined and, as the Cayley transform of a strict contraction, has positive semidefinite real part, and so is $f(\mathbf{C})$. Clearly, since $V$ is an isometry, $f_\emptyset=I_\mathcal{Y}$.

Let us prove the converse.
The formal power series 
\begin{equation}\label{cayley}
F(z):=(f(z)-I_\mathcal{Y})(f(z)+I_\mathcal{Y})^{-1}
\end{equation}
is well defined since $f_\emptyset+I_\mathcal{Y}=2I_\mathcal{Y}$ is invertible and so is $f(z)+I_\mathcal{Y}$. Moreover, $F$ belongs to the \emph{non-commutative Schur--Agler class $\mathcal{SA}_N^{\rm nc}(\mathcal{Y})$}, i.e., the formal power series  $F$ evaluated at any $\mathbf{C}\in\mathcal{D}^N$ is well defined and $\| F(\mathbf{C})\|\leq 1$. By \cite{BGM}, there exists a Hilbert space $\mathcal{H}$, a resolution of identity $\mathbf{P}=(P_1,\ldots,P_N)\in\mathcal{L(H)}^N$, and a unitary operator {\small $U=\left[\begin{array}{cc} A & B\\ C & D\end{array}\right]\in\mathcal{L(H\oplus Y)}$} such that
\begin{equation}\label{sa-nc}
F(z)=D+C(I_\mathcal{Y}-(z\mathbf{P})A)^{-1}(z\mathbf{P})B
\end{equation}
where we set $z\mathbf{P}:=\sum_{k=1}^Nz_kP_k$. To get the representation \eqref{ha-nc} for $f$, we will apply a trick which is well known in one-variable system theory (see, e.g., \cite{Ar}). Consider a \emph{non-commutative linear system}
$\Sigma =(N; U; \mathbf{P}; \mathcal{H,Y})$, i.e., a system of equations
\begin{equation}\label{scat}
\left\{ \begin{array}{lll}
x(z) &=& (z\mathbf{P})Ax(z)+(z\mathbf{P})Bu(z),\\
y(z) &=& Cx(z)+Du(z),
\end{array}\right.
\end{equation}
where $x(z)\in\mathcal{L(H)}\left\langle \left\langle z_1,\ldots,z_N\right\rangle\right\rangle,\ u(z),y(z)\in\mathcal{L(Y)}\left\langle \left\langle z_1,\ldots,z_N\right\rangle\right\rangle $ (the corresponding system of equations for coefficients of these formal power series is one of the systems with evolution on the free semigroup considered in \cite{BGM}). The system \eqref{scat} is equivalent to the system
\begin{equation}\label{scat'}
\left\{ \begin{array}{lll}
x(z) &=& (I_\mathcal{H}-(z\mathbf{P})A)^{-1}(z\mathbf{P})Bu(z),\\
y(z) &=& F(z)u(z).
\end{array}\right.
\end{equation}
The second equation means that $F$ is the \emph{transfer function} of the system $\Sigma$, or $\Sigma$ is a \emph{realization} of the formal power series $F$. Let us find a realization $\widetilde{\Sigma} =(N; \widetilde{U}; \mathbf{P}; \mathcal{H,Y})$ of $f$. To this end (now the trick appears!) we apply the so-called \emph{diagonal transform}:
\begin{equation}\label{diag}
u(z)=\widetilde{y}(z)+\widetilde{u}(z),\qquad y(z)=\widetilde{y}(z)-\widetilde{u}(z).
\end{equation}
Then we get $\widetilde{y}(z)=f(z)\widetilde{u}(z)$, i.e., an analogue of the second equation in \eqref{scat'}. Suppose that the operator $I_\mathcal{Y}-D$ is invertible. Then an easy calculation gives the desired system realization $\widetilde{\Sigma}$:
\begin{equation}\label{realiz}
\left\{ \begin{array}{lll}
x(z) &=& (z\mathbf{P})\widetilde{A}x(z)+(z\mathbf{P})\widetilde{B}u(z),\\
y(z) &=& \widetilde{C}x(z)+\widetilde{D}u(z),
\end{array}\right.
\end{equation}
where
\begin{eqnarray*}
\widetilde{A}=A+B(I_\mathcal{Y}-D)^{-1}C, & \widetilde{B}=2B(I_\mathcal{Y}-D)^{-1}\\
\widetilde{C}=(I_\mathcal{Y}-D)^{-1}C,   & \widetilde{D}=(I_\mathcal{Y}-D)^{-1}(I_\mathcal{Y}+D).
\end{eqnarray*}
In our case $\widetilde{D}=f_\emptyset=I_\mathcal{Y}$ and $D=F_\emptyset=0$. Then
$$
\widetilde{A}=A+BC, \quad \widetilde{B}=2B,\quad
\widetilde{C}=C.
$$
Moreover, since $U$ is a unitary operator, in this case $B$ is an isometry, $C$ is a coisometry, $A+BC$ is unitary, and $A^*B=0,\ AC^*=0$. Thus, we may write
\begin{eqnarray*}
f(z) &=& I_\mathcal{Y}+2C(I_\mathcal{H}-(z\mathbf{P})(A+BC))^{-1}(z\mathbf{P})B\\
     &=& I_\mathcal{Y}+2C(I_\mathcal{H}-(z\mathbf{P})(A+BC))^{-1}(z\mathbf{P})(A+BC)(A+BC)^*B\\
     &=& I_\mathcal{Y}+2C(I_\mathcal{H}-(z\mathbf{P})(A+BC))^{-1}(z\mathbf{P})(A+BC)C^*\\
     &=& C(I_\mathcal{H}+(z\mathbf{P})(A+BC))(I_\mathcal{H}-(z\mathbf{P})(A+BC))^{-1}C^*\\
     &=& V^*(I_\mathcal{H}+z\mathbf{G})(I_\mathcal{H}-z\mathbf{G})^{-1}V,
\end{eqnarray*}
where $G_k=P_k(A+BC),\ k=1,\ldots,N$ and $V=C^*$. By Proposition~\ref{prop:gn}, $\mathbf{G}\in\mathcal{G}_N$. Since $C$ is a coisometry, $V=C^*$ is an isometry. Thus we have obtained a representation \eqref{ha-nc} of $f$.
\end{proof}
\begin{cor}\label{cor:coef}
A formal power series $f\in\mathcal{L(Y)}\left\langle \left\langle z_1,\ldots,z_N\right\rangle\right\rangle$ belongs to the class $\mathcal{HA}_N^{nc}(\mathcal{Y})$ and satisfies $f_\emptyset=\frac{I_\mathcal{Y}}{2}$ if and only if there exist a Hilbert space $\mathcal{H}$, an $N$-tuple of operators $\mathbf{G}\in\mathcal{G}_N\cap\mathcal{L(H)}^N$, and an isometry $V\in\mathcal{L(Y,H)}$ such that the sequence 
$$
a_\emptyset:=I_\mathcal{Y},\ a_w:=f_w \qquad {\rm for}\quad w\in\mathcal{F}_N\setminus\{\emptyset\}, 
$$
satisfies
$$
a_w=V^*\mathbf{G}^wV,\qquad w\in\mathcal{F}_N. 
$$
\end{cor}
\begin{proof}
The statement follows from the representation \eqref{ha-nc} for $\widetilde{f}=2f$:
\begin{eqnarray*}
\widetilde{f}(z) &=& V^*(2(I_\mathcal{H}-z\mathbf{G})^{-1}-I_\mathcal{H})V=V^*\left(2\sum_{j=0}^\infty (z\mathbf{G})^j-I_\mathcal{H}\right)V \\
              &=& I_\mathcal{H}+2V^*\left(\sum_{w\in\mathcal{F}_N\setminus\{\emptyset\}}\mathbf{G}^wz^w\right)V.
\end{eqnarray*}
\end{proof}
\begin{rem}\label{rem:mpd}
In \cite{AK2} it has been shown that a formal power series $F$ belongs to the class $\mathcal{SA}_N^{\rm nc}(\mathcal{Y})$ (or more generally, to the class $\mathcal{SA}_N^{\rm nc}(\mathcal{U,Y})$) if and only if the series for $F(\mathbf{C})$ converges to a contractive operator for every $\mathbf{C}\in\mathcal{D}^N_{\rm matr}$, i.e., it is enough to test values of $F$ on $N$-tuples of strictly contractive matrices of same size $n\times n,\ n=1,2,\ldots$. The analogous statement for the class $\mathcal{HA}_N^{\rm nc}(\mathcal{Y})$ is true, too: a formal power series $f$ belongs to the class $\mathcal{HA}_N^{\rm nc}(\mathcal{Y})$ if and only if the series for $f(\mathbf{C})$ converges and $\Re f(\mathbf{C})$ is positive semidefinite for every $\mathbf{C}\in\mathcal{D}^N_{\rm matr}$. Indeed, this follows from the fact that the Cayley transform $f\mapsto F$ defined by \eqref{cayley} is an injection from $\mathcal{HA}_N^{\rm nc}(\mathcal{Y})$ into $\mathcal{SA}_N^{\rm nc}(\mathcal{Y})$. 
\end{rem}

\section{The Carath\'{e}odory interpolation problem}\label{sec:prob}
In this section we will consider $\mathcal{F}_N$ as a sub-semigroup of the \emph{free semigroup with involution} $\hat{\mathcal{F}}_{2N}$. The latter is the free semigroup $\mathcal{F}_{2N}$ with the generators $g_1,\ldots,g_N,g_{N+1},\ldots,g_{2N}$ and the neutral element $\emptyset$, endowed with the involution ``$*$" defined as follows: $g_k^*:=g_{k+N}$ for $k=1,\ldots,N$, $g_k^*:=g_{k-N}$ for $k=N+1,\ldots,2N$, $\emptyset^*:=\emptyset$, and $(g_{i_1}\cdots g_{i_l})^*:=g_{i_l}^*\cdots g_{i_1}^*$ for every $l\in\mathbb{N}$ and $i_j\in\{1,\ldots,2N\},\ j=1,\ldots,l$. For a set $\Omega\subset\hat{\mathcal{F}}_{2N}$ we define the set $\Omega^*:=\{ w\in\hat{\mathcal{F}}_{2N}:\ w^*\in\Omega\}.$ Let us introduce also the unital $*$-algebra $\mathcal{A}_N(\mathcal{Y})$ as the algebra $\mathcal{L(Y)}\left\langle z_1,\ldots,z_N,z_{N+1},\ldots z_{2N}\right\rangle$  endowed with the involution ``$*$" defined as follows: 1) $z_k^*:=z_{k+N}$ for $k=1,\ldots,N$, $z_k^*:=z_{k-N}$ for $k=N+1,\ldots,2N$, $(z_{i_1}\cdots z_{i_l})^*:=z_{i_l}^*\cdots z_{i_1}^*$ for every $l\in\mathbb{N}$ and $i_j\in\{1,\ldots,2N\},\ j=1,\ldots,l$, thus for $\hat{z}:=(z_1,\ldots,z_N,z_{N+1},\ldots z_{2N})=(z_1,\ldots,z_N,z_1^*,\ldots,z_N^*)$ and $w\in\hat{\mathcal{F}}_{2N}$ one has $(\hat{z}^w)^*=\hat{z}^{w^*}$; 2) for arbitrary finite set $\Omega\subset\hat{\mathcal{F}}_{2N}$ and a polynomial $p(\hat{z})=\sum_{w\in\Omega}p_w\hat{z}^w$, $$p(\hat{z})^*=\left(\sum_{w\in\Omega}p_w\hat{z}^w\right)^*:=\sum_{w\in\Omega}p_w^*\hat{z}^{w^*}=\sum_{w\in\Omega^*}p_{w^*}^*\hat{z}^w$$
(here $p_w^*$ is the adjoint operator to $p_w$ in $\mathcal{L(Y)}$). 

A finite set $\Lambda\subset\mathcal{F}_N$ will be called \emph{admissible} if $g_kw\in\mathcal{F}_N\setminus\Lambda$ and $wg_k\in\mathcal{F}_N\setminus\Lambda$ for every $w\in\mathcal{F}_N\setminus\Lambda$ and $k=1,\ldots,N$. Clearly, if the set $\Lambda$ is admissible and non-empty then $\emptyset\in\Lambda$, and if $\Lambda$ is admissible, non-empty and $\Lambda\neq\{\emptyset\}$ then there is a $k\in\{ 1,\ldots,N\}$ such that $g_k\in\Lambda$. For example, the set $\Lambda_m:=\{ w\in\mathcal{F}_N:\ |w|\leq m\}$ is admissible.

Let us pose now the \emph{Carath\'{e}odory interpolation problem in the class $\mathcal{HA}_N^{\rm nc}(\mathcal{Y})$}.
\begin{prob}\label{prob:c-gen}
Let $\Lambda\subset\mathcal{F}_N$ be an admissible set. Given a collection of operators $\{ c_w\}_{w\in\Lambda}\in\mathcal{L(Y)}$, with $c_\emptyset\geq 0$, find 
$f\in\mathcal{HA}_N^{\rm nc}(\mathcal{Y})$ such that
$$
f_\emptyset=\frac{c_\emptyset}{2},\quad f_w=c_w\quad {\rm for}\ w\in\Lambda\setminus\{\emptyset\}.
$$
\end{prob}
We will start with the special case of this problem where $c_\emptyset=I_\mathcal{Y}$.
\begin{prob}\label{prob:c}
Let $\Lambda\subset\mathcal{F}_N$ be an admissible set. Given a collection of operators $\{ c_w\}_{w\in\Lambda}\in\mathcal{L(Y)}$, with $c_\emptyset=I_\mathcal{Y}$, find 
$f\in\mathcal{HA}_N^{\rm nc}(\mathcal{Y})$ such that
$$f_\emptyset=\frac{c_\emptyset}{2}=\frac{I_\mathcal{Y}}{2},\quad f_w=c_w\quad {\rm for}\ w\in\Lambda\setminus\{\emptyset\}.
$$
\end{prob}
 
From Corollary~\ref{cor:coef} we obtain the following result.
\begin{thm}\label{thm:g}
Problem~\ref{prob:c} has a solution if and only if there exist a Hilbert space $\mathcal{H}$, an $N$-tuple of operators $\mathbf{G}\in\mathcal{G}_N\cap\mathcal{L(H)}^N$, and an isometry $V\in\mathcal{L(Y,H)}$ such that
\begin{equation}\label{mom-nc}
c_w=V^*\mathbf{G}^wV,\qquad w\in\Lambda.
\end{equation}
\end{thm}
\begin{rem}\label{rem:arb}
Theorem~\ref{thm:g} holds true for the case where Problem~\ref{prob:c} is formulated for an arbitrary set $\Lambda\subset\mathcal{F}_N$, not necessarily finite and admissible.
\end{rem}
Let us note that \eqref{mom-nc} is a non-commutative multivariable counterpart of \eqref{mom-o}. Theorem~\ref{thm:g} gives a criterion on solvability of Problem~\ref{prob:c} in the ``existence terms". We are going to obtain also another criterion, in terms of positivity of certain non-commutative polynomial whose coefficients are determined by the problem data. 

Let $\mathcal{E}$ be a Hilbert space, and $\mathbf{T}=(T_1,\ldots,T_N)\in\mathcal{L(E)}^N$. Then set $$\hat{\mathbf{T}}:=(T_1,\ldots,T_N,T_1^*,\ldots,T_N^*)\in\mathcal{L(E)}^{2N}.$$ For a finite set $\Omega\subset\hat{\mathcal{F}}_{2N}$ and a polynomial $p(\hat{z})=\sum_{w\in\Omega}p_w\hat{z}^w\in\mathcal{A}_N(\mathcal{Y})$ define $$p(\mathbf{T})=p(\hat{\mathbf{T}}):=\sum_{w\in\Omega}p_w\otimes\hat{\mathbf{T}}^w\in\mathcal{L(Y\otimes E)}.$$
In particular, if $\Omega=\Lambda\cup\Lambda^*$ where $\Lambda\subset\mathcal{F}_N$ is a finite set, and $$p(\hat{z})=\sum_{w\in\Lambda\cup\Lambda^*}p_w\hat{z}^w=p_\emptyset+\sum_{w\in\Lambda\setminus\{\emptyset\}}p_wz^w+\sum_{w\in\Lambda^*\setminus\{\emptyset\}}p_wz^{*w},$$ we have 
$$p(\mathbf{T})=\sum_{w\in\Lambda\cup\Lambda^*}p_w\otimes\hat{\mathbf{T}}^w=p_\emptyset\otimes I_\mathcal{E}+\sum_{w\in\Lambda\setminus\{\emptyset\}}p_w\otimes \mathbf{T}^w+\sum_{w\in\Lambda^*\setminus\{\emptyset\}}p_w\otimes\mathbf{T}^{*w},$$
where $z^*:=(z_1^*,\ldots,z_N^*)$ and $\mathbf{T}^*:=(T_1^*,\ldots,T_N^*)$, and one identifies $z^w=\hat{z}^w,\ \mathbf{T}^w=\hat{\mathbf{T}}^w$ for $w\in\mathcal{F}_N\subset\hat{\mathcal{F}}_{2N}$, and $z^{*w}=\hat{z}^{*w},\ \mathbf{T}^{*w}=\hat{\mathbf{T}}^{*w}$ for $w\in\mathcal{F}_N^*\subset\hat{\mathcal{F}}_{2N}$.
Thus, the evaluation of polynomials from $\mathcal{A}_N(\mathcal{Y})$ on $N$-tuples of operators is well defined.  
\begin{lem}\label{lem:main}
Let $\emptyset\in\Lambda\subset\mathcal{F}_N$ be a finite set.
\begin{description}
	\item[I] A polynomial $$p(\hat{z})=I_\mathcal{Y}+\sum_{w\in\Lambda\setminus\{\emptyset\}}p_wz^w+\sum_{w\in\Lambda^*\setminus\{\emptyset\}}p_wz^{*w}\in\mathcal{A}_N(\mathcal{Y})$$ is positive semidefinite on $\mathcal{U}^N$ if and only if there exist a Hilbert space $\mathcal{H}$, an $N$-tuple of operators $\mathbf{G}\in\mathcal{G}_N\cap\mathcal{L(H)}^N$, and an isometry $V\in\mathcal{L(Y,H)}$ such that 
	\begin{eqnarray}
	p_w=p_{w^*}^* &=& V^*\mathbf{G}^wV,\qquad w\in\Lambda,\label{I1}\\
	0 &=& V^*\mathbf{G}^wV,\qquad w\in \mathcal{F}_N\setminus \Lambda.\label{I2}
\end{eqnarray}	
\item[II] A polynomial $$p(\hat{z})=I_\mathcal{Y}+\sum_{w\in\Lambda\setminus\{\emptyset\}}p_wz^w+\sum_{w\in\Lambda^*\setminus\{\emptyset\}}p_wz^{*w}\in\mathcal{A}_N(\mathcal{Y})$$ is positive semidefinite on $\mathcal{G}^N$ if and only if there exist a Hilbert space $\mathcal{K}$, an $N$-tuple of operators $\mathbf{U}\in\mathcal{U}^N\cap\mathcal{L(K)}^N$, and an isometry $W\in\mathcal{L(Y,K)}$ such that 
	\begin{eqnarray}
	p_w=p_{w^*}^* &=& W^*\mathbf{U}^wW,\qquad w\in\Lambda,\label{II1}\\
	0 &=& W^*\mathbf{U}^wW,\qquad w\in \mathcal{F}_N\setminus \Lambda.\label{II2}
\end{eqnarray}
\end{description}
\end{lem}
\begin{proof}
I. If the polynomial $p$ is positive semidefinite on $\mathcal{U}^N$ then $p_w=p_{w^*}^*$ for $w\in\Lambda$. This can be seen from a McCullough factorization \cite{McC3}: $p(\hat{z})=h(z)^*h(z)$, where $h(z)=\sum_{w\in\mathcal{F}_N:\,|w|\leq m}h_wz^w\in\mathcal{L(Y,V)}\left\langle z_1,\ldots,z_N\right\rangle$, with an auxiliary Hilbert space $\mathcal{V}$. Set $$f(z):=\frac{I_\mathcal{Y}}{2}+\sum_{w\in\Lambda\setminus\{\emptyset\}}p_wz^w.$$
Then $p(\mathbf{C})=2\Re f(\mathbf{C})\geq 0$ for every $\mathbf{C}\in\mathcal{D}^N\cap\mathcal{L(E)}^N$, with a Hilbert space $\mathcal{E}$. Indeed, since $\mathbf{C}$ has a unitary dilation (see \cite{SzNF}) $\mathbf{U}\in\mathcal{U}^N\cap\mathcal{L(K)}^N$, i.e., $\mathcal{K}\supset\mathcal{E}$ and
$$\mathbf{C}^w=P_\mathcal{E}\mathbf{U}^w\big|_{\mathcal{E}},\qquad w\in\mathcal{F}_N,$$
and $p(\mathbf{U})=2\Re f(\mathbf{U})\geq 0$, we get
$$p(\mathbf{C})=(I_\mathcal{Y}\otimes P_\mathcal{E})p(\mathbf{U})\big|_\mathcal{Y\otimes E}\geq 0.$$ 
Thus, $f\in\mathcal{HA}^{\rm nc}_N(\mathcal{Y})$ and $f_\emptyset = \frac{I_\mathcal{Y}}{2}$. By Corollary~\ref{cor:coef}, there exists a representation \eqref{I1}--\eqref{I2}.

Conversely, if $p$ has a representation \eqref{I1}--\eqref{I2} then $p(\hat{z})= f(z)+f(z)^*$, where $$f(z)=\frac{I_\mathcal{Y}}{2}+\sum_{w\in\Lambda\setminus\{\emptyset\}}p_wz^w=\frac{I_\mathcal{Y}}{2}+\sum_{w\in\Lambda\setminus\{\emptyset\}}V^*\mathbf{G}^wVz^w.$$
By Corollary~\ref{cor:coef}, $f\in\mathcal{HA}^{\rm nc}_N(\mathcal{Y})$. Hence, $$p(\mathbf{U})=2\Re f(\mathbf{U})=\lim_{r\uparrow 1}2\Re f(r\mathbf{U})\geq 0,\qquad \mathbf{U}\in\mathcal{U}^N.$$

II. Let $p$ be positive semidefinite on $\mathcal{G}_N$. Let $\mathcal{A}_{\mathcal{U}^N}$ be the $C^*$-algebra obtained as  the norm completion of the quotient of unital $*$-algebra $\mathcal{A}_N=\mathcal{A}_N(\mathbb{C})$ with the seminorm
$$\| q\| :=\sup_{\mathbf{U}\in\mathcal{U}^N}\| q(\mathbf{U})\|=\sup_{\mathbf{U}\in\mathcal{U}^N}\| q(U_1,\ldots,U_N,U_1^*,\ldots,U_N^*)\|,$$
by the two-sided ideal of elements of zero seminorm. 

Let us show that the restriction of the quotient map above to the subspace $\mathcal{B}_N\subset\mathcal{A}_N$ of polynomials of the form 
\begin{equation}\label{q}
q(\hat{z})=q_\emptyset +\sum_{w\in\mathcal{F}_N:\,0<|w|\leq m}q_wz^w+\sum_{w\in\mathcal{F}_N^*:\,0<|w|\leq m}q_wz^{*w}
\end{equation}
 is injective, i.e., that if $q\in\mathcal{B}_N$ is non-zero then the corresponding coset $[q]\in\mathcal{A}_{\mathcal{U}^N}$ is non-zero. Indeed, if $[q]=[0]$ then $q(\mathbf{U})=0$ for every $\mathbf{U}\in\mathcal{U}^N$. In particular, $q$ is positive semidefinite on $\mathcal{U}^N$. By \cite{McC3}, there exists a polynomial $h(z)=\sum_{w\in\mathcal{F}_N:\,|w|\leq m}h_wz^w\in\mathcal{L}(\mathbb{C},\mathbb{C}^r)\left\langle z_1,\ldots,z_N\right\rangle$, with some $r\in\mathbb{N}$, such that $q(\hat{z})=h(z)^*h(z)$. Then 
$h$ vanishes on $\mathcal{U}^N$. In particular, for every $n\in\mathbb{N}$ the polynomial $h$ vanishes on $\mathcal{U}^N\cap (\mathbb{C}^{n\times n})^N$. The latter set is the uniqueness set for functions of matrix entries, which are holomorphic on a domain containing this set (see, e.g., \cite{Sh}). Then $h$ vanishes on the all of $(\mathbb{C}^{n\times n})^N$, for each $n\in\mathbb{N}$. By the Amitsur--Levitzki theorem (see \cite[pp. 22-23]{Ro}) such a polynomial should be zero, i.e., $h_w=0$ for all $w\in\mathcal{F}_N:\,|w|\leq m$. Then $q_w=0$ for all $w\in\mathcal{F}_N\cup\mathcal{F}_N^*:\,|w|\leq m$. 

Denote by $\mathcal{B}_{\mathcal{U}^N}$ the image of the subspace $\mathcal{B}_N$ under the quotient map above. This subspace of the $C^*$-algebra $\mathcal{A}_{\mathcal{U}^N}$ is \emph{selfadjoint}, i.e., $$\mathcal{B}_{\mathcal{U}^N}^*:=\{ [q]^*=[q^*]:\ [q]\in\mathcal{B}_{\mathcal{U}^N}\}=\mathcal{B}_{\mathcal{U}^N}.$$ Define the linear map $\varphi:\ \mathcal{B}_{\mathcal{U}^N}\to\mathcal{L(Y)}$ by  $\varphi([z^w])=p_w$ for $w\in\Lambda$, $\varphi([z^{*w}])=p_w$ for $w\in\Lambda^*$, and $\varphi([z^w])=\varphi([z^{*w^*}])=0$ for $w\in\mathcal{F}_N\setminus\Lambda$. By the result of the previous paragraph together with the Amitsur--Levitzki theorem mentioned there, this linear map is correctly defined. Let us show that $\varphi$ is \emph{completely positive}, i.e., that for every $n\in\mathbb{N}$ the map $$\varphi_n:= {\rm id}_n\otimes\varphi:\ \mathbb{C}^{n\times n} \otimes\mathcal{B}_{\mathcal{U}^N}\to\mathbb{C}^{n\times n}\otimes\mathcal{L(Y)}$$ (here ${\rm id}_n$ is the identity map from the $C^*$-algebra $\mathbb{C}^{n\times n}$ onto itself) is \emph{positive}. The latter means, in turn, that $\varphi_n$ maps positive elements (in the sense  of the $C^*$-algebra $\mathbb{C}^{n\times n}\otimes\mathcal{A}_{\mathcal{U}^N}$) from $\mathbb{C}^{n\times n}\otimes\mathcal{B}_{\mathcal{U}^N}$ into positive elements in the $C^*$-algebra $\mathbb{C}^{n\times n}\otimes\mathcal{L(Y)}$. Let $[q]\in\mathbb{C}^{n\times n}\otimes\mathcal{B}_{\mathcal{U}^N}$ be a positive element of the $C^*$-algebra $\mathbb{C}^{n\times n}\otimes\mathcal{A}_{\mathcal{U}^N}$, i.e.,
$[q]=[h]^*[h]$ with some $[h]\in\mathbb{C}^{n\times n}\otimes\mathcal{A}_{\mathcal{U}^N}$. One can think of $[q]$ as of the $n\times n$ matrix $([q]_{ij})_{i,j=1,\ldots,n}$ whose entries $[q]_{ij}=[q_{ij}]\in\mathcal{B}_{\mathcal{U}^N}$ and $q_{ij}\in\mathcal{B}_N$, and thus $q\in\mathbb{C}^{n\times n}\otimes\mathcal{B}_N$ is a polynomial of the form \eqref{q} with the coefficients from $\mathbb{C}^{n\times n}$. Let us observe that by virtue of the definition of the $C^*$-algebra $\mathcal{A}_{\mathcal{U}^N}$, for an arbitrary $[x]\in\mathcal{A}_{\mathcal{U}^N}$ its values on $\mathcal{U}^N$ are well defined. In particular, if $x\in\mathcal{B}_N$ then $[x](\mathbf{U})=x(\mathbf{U})$ for any $\mathbf{U}\in\mathcal{U}^N$. Therefore, for an arbitrary $[x]=([x]_{ij})_{i,j=1,\ldots,n}=([x_{ij}])_{i,j=1,\ldots,n}\in\mathbb{C}^{n\times n}\otimes\mathcal{A}_{\mathcal{U}^N}$ one defines correctly $[x](\mathbf{U}):=([x_{ij}](\mathbf{U}))_{i,j=1,\ldots,n},\ \mathbf{U}\in\mathcal{U}^N$. In particular, if $x=(x_{ij})_{i,j=1,\ldots,n}\in\mathbb{C}^{n\times n}\otimes\mathcal{B}_N$ then $[x](\mathbf{U})=x(\mathbf{U})=(x_{ij}(\mathbf{U}))_{i,j=1,\ldots,n}$ for any $\mathbf{U}\in\mathcal{U}^N$. Since $q(\mathbf{U})=[q](\mathbf{U})=[h](\mathbf{U})^*[h](\mathbf{U})$ is positive semidefinite for every  $\mathbf{U}\in\mathcal{U}^N$, it follows that the polynomial $q$ (with the coefficients from $\mathbb{C}^{n\times n}$) is positive semidefinite on $\mathcal{U}^N$. From the McCullough factorization theorem \cite{McC3} we deduce that $q_\emptyset\geq 0$. If $q_\emptyset=I_n$, then by Part I of this Lemma, there exist a Hilbert space $\mathcal{H}$, an $N$-tuple of operators $\mathbf{G}\in\mathcal{G}_N\cap\mathcal{L}(\mathcal{H})^N$, and an isometry $V\in\mathcal{L}(\mathbb{C}^n,\mathcal{H})$ such that 
\begin{eqnarray*}
q_w=q_{w^*}^* &=& V^*\mathbf{G}^wV,\qquad w\in\mathcal{F}_N:\,|w|\leq m,\\
0   &=& V^*\mathbf{G}^wV,\qquad w\in\mathcal{F}_N:\,|w|>m.
\end{eqnarray*}
Then we have
\allowdisplaybreaks
\begin{eqnarray*}
\varphi_n([q]) &=& ({\rm id}_n\otimes \varphi)\left(I_n\otimes [1]+\sum_{w\in\mathcal{F}_N:\,0<|w|\leq m}q_w\otimes [z^w]\right.\\
&+& \left.\sum_{w\in\mathcal{F}_N^*:\,0<|w|\leq m}q_w\otimes [z^{*w}] \right)\\ 
&=& I_{\mathbb{C}^n\otimes \mathcal{Y}}+\sum_{w\in\Lambda\setminus\{\emptyset\} }V^*\mathbf{G}^wV\otimes p_w+ \sum_{w\in\Lambda^*\setminus\{\emptyset\} }V^*\mathbf{G}^{*w}V\otimes p_w\\
&=& (V^*\otimes I_\mathcal{Y})p^{\rm l}(\mathbf{G})(V\otimes I_\mathcal{Y})\geq 0
\end{eqnarray*}
(positivity in the $C^*$-algebra $\mathbb{C}^{n\times n}\otimes\mathcal{L(Y)}\cong\mathcal{L}(\mathbb{C}^n\otimes\mathcal{Y})$ is operator positive semidefiniteness). 
In the case where $q_\emptyset >0$ we  define $\tilde{q}(\hat{z}):=q_\emptyset^{-1/2}q(\hat{z})q_\emptyset^{-1/2}$. Since $\varphi_n([\tilde{q}])\geq 0$, we get $$\varphi_n([q])=(q_\emptyset^{1/2}\otimes I_\mathcal{Y})\varphi_n([\tilde{q}])(q_\emptyset^{1/2}\otimes I_\mathcal{Y})\geq 0.$$
In the case where the matrix $q_\emptyset$ is degenerate we set $q_\epsilon(\hat{z}) :=\epsilon I_n+q(\hat{z})$ for  $\epsilon >0$. Then $q_\epsilon$ is positive definite on $\mathcal{U}^N$ and ${(q_\epsilon)}_\emptyset=\epsilon I_n+q_\emptyset >0$. Since $\varphi_n([q_\epsilon])\geq 0$, we get
 $$
\varphi_n([q]) = \lim_{\epsilon\downarrow 0}\varphi_n([q_\epsilon ])\geq 0.
$$
Finally, we have obtained that $\varphi: \mathcal{B}_{\mathcal{U}^N}\rightarrow \mathcal{L(Y)}$ is completely positive.

Since we have $\varphi ([1])=I_\mathcal{Y}$, by the Arveson extension theorem \cite{Arv1} there exists a completely positive map $\widetilde{\varphi}:\mathcal{A}_{\mathcal{U}^N}\rightarrow \mathcal{L(Y)}$ which extends $\varphi$. By the Stinespring theorem \cite{St}, there exists a $*$-representation $\pi$ of $\mathcal{A}_{\mathcal{U}^N}$ in some Hilbert space $\mathcal{K}$ and an isometry $W\in\mathcal{L(Y,K)}$ such that
$$\widetilde{\varphi}(a)=W^*\pi(a)W,\qquad a\in\mathcal{A}_{\mathcal{U}^N}.$$
In particular, we get
\begin{eqnarray}
p_w &=& \varphi([z^w])=\widetilde{\varphi}([z^w])=W^*\pi([z^w])W=W^*\mathbf{U}^wW,\quad w\in\Lambda,\label{II1'}\\
p_w &=& \varphi([z^{*w}])=\widetilde{\varphi}([z^{*w}])=W^*\pi([z^{*w}])W=W^*\mathbf{U}^{*w}W,\quad w\in\Lambda^*,\label{II2'}\\
0   &=& \varphi([z^w])=\widetilde{\varphi}([z^w])=W^*\pi([z^w])W=W^*\mathbf{U}^wW,\quad w\in \mathcal{F}_N\setminus \Lambda,
\label{II3'}
\end{eqnarray}
where we set $\mathbf{U}:=(\pi([z_1]),\ldots,\pi([z_N]))$. We have $\mathbf{U}\in\mathcal{U}^N$. Indeed, since $$\| [1-z_k^*z_k]\|_{\mathcal{A}_{\mathcal{U}^N}}=\| [1-z_kz_k^*]\|_{\mathcal{A}_{\mathcal{U}^N}}=0,$$ we get $[z_k^*z_k]=[z_kz_k^*]=[1]$. Hence
\begin{eqnarray*} 
U_k^*U_k &=& \pi([z_k])^*\pi([z_k])=\pi([z_k^*z_k])=\pi([1])=I_\mathcal{K},\quad k=1,\ldots,N,\\ U_kU_k^* &=& \pi([z_k])\pi([z_k])^*=\pi([z_kz_k^*])=\pi([1])=I_\mathcal{K},\quad k=1,\ldots,N.
\end{eqnarray*}
Clearly, \eqref{II1'} and \eqref{II2'} imply that $p_w=p_{w^*}^*$ for $w\in\Lambda$.
Thus, representation \eqref{II1'}--\eqref{II3'} for the coefficients of $p$ is a desired representation \eqref{II1}--\eqref{II2}.

Conversely, if the coefficients of $p$ have a representation \eqref{II1}--\eqref{II2} then for any $\mathbf{G}\in\mathcal{G}_N\cap\mathcal{L}(\mathcal{H})^N$ one has
\begin{eqnarray*}
p(\mathbf{G}) &=& I_\mathcal{Y\otimes H}+ \sum_{w\in\Lambda\setminus\{\emptyset\} }p_w\otimes\mathbf{G}^w+\sum_{w\in\Lambda^*\setminus\{\emptyset\}}p_w\otimes\mathbf{G}^{*w}\\
&=& I_\mathcal{Y\otimes H}+ \sum_{w\in\Lambda\setminus\{\emptyset\}}W^*\mathbf{U}^wW\otimes\mathbf{G}^w+\sum_{w\in\Lambda^*\setminus\{\emptyset\}}W^*\mathbf{U}^{*w}W\otimes\mathbf{G}^{*w}\\
&=& I_\mathcal{Y\otimes H}+ \sum_{w\in\mathcal{F}_N\setminus\{\emptyset\}}W^*(\mathbf{U})^wW\otimes\mathbf{G}^w+
\sum_{w\in\mathcal{F}_N^*\setminus\{\emptyset\}}W^*\mathbf{U}^{*w}W\otimes\mathbf{G}^{*w}
\end{eqnarray*}
(note that the sums are finite!). A formal power series $$f(z):=\frac{I_\mathcal{H}}{2}+\sum_{w\in\mathcal{F}_N\setminus\{\emptyset\}}\mathbf{G}^wz^w$$
by Corollary~\ref{cor:coef} belongs to the class $\mathcal{HA}^{\rm nc}_N(\mathcal{H})$. Therefore, for $0<r<1$ one has
\begin{multline*}
I_\mathcal{Y\otimes H}+ \sum_{w\in\mathcal{F}_N\setminus\{\emptyset\}}W^*(r\mathbf{U})^wW\otimes\mathbf{G}^w+
\sum_{w\in\mathcal{F}_N^*\setminus\{\emptyset\}}W^*(r\mathbf{U})^{*w}W\otimes\mathbf{G}^{*w}\\
= (W^*\otimes I_\mathcal{H})2\Re f^{\rm l}(r\mathbf{U})(W\otimes I_\mathcal{H})\geq 0.
\end{multline*}
The sum on the left is finite. Hence, by letting $r\uparrow 1$, we get $p(\mathbf{G})\geq 0$. Thus, $p$ is positive semidefinite on $\mathcal{G}_N$.
\end{proof}
Let us introduce the class ${\rm Nilp}_N$ of $N$-tuples $\mathbf{T}=(T_1,\ldots,T_N)$ of \emph{jointly nilpotent} bounded linear operators on a common Hilbert space, i.e., such that for some $r\in\mathbb{N}$ one has
$$\mathbf{T}^w=0\quad {\rm for\ all}\ w\in\mathcal{F}_N:\ |w|\geq r.$$ 
The minimal such $r$ is called the \emph{rank of joint nilpotency}. Let $\Lambda\subset\mathcal{F}_N$ be an admissible set. We shall say that $\mathbf{T}=(T_1,\ldots,T_N)\in {\rm Nilp}_N$ is an $N$-tuple of \emph{$\Lambda$-jointly nilpotent} operators if 
$$ \mathbf{T}^w=0\quad {\rm for\ all}\ w\in\mathcal{F}_N\setminus\Lambda.$$ 
In this case the rank of joint nilpotency of $\mathbf{T}$ is at most $\max_{w\in\Lambda}|w|+1$. We denote the class of $N$-tuples of $\Lambda$-jointly nilpotent operators by ${\rm Nilp}_N(\Lambda)$. For  $\Lambda_m:=\{ w\in\mathcal{F}_N:\ |w|\leq m\}$, $N$-tuples of $\Lambda_m$-jointly nilpotent operators are exactly those whose rank of joint nilpotency is at most $m+1$. 
\begin{ex}\label{ex:shifts}
Let $\Lambda\subset\mathcal{F}_N$ be an admissible set. Let $\mathcal{H}_\Lambda$ be a finite-dimensional Hilbert space whose orthonormal basis is identified with the set $\Lambda$. For $k=1,\ldots,N$ define the non-commutative backward shifts $S_k\in\mathcal{L}(\mathcal{H}_\Lambda )$ by their action on basis vectors:
$$S_kw =\left\{ \begin{array}{ll}
w' & {\rm if\ } w=g_kw'\ {\rm with\ some\ } w'\in\mathcal{F}_N,\\
0  & {\rm otherwise}.
\end{array}
\right. $$
Since $\Lambda$ is admissible, these operators are correctly defined (if $w=g_kw'$ and $w\in\Lambda$ then $w'\in\Lambda$). The $N$-tuple $\mathbf{S}:=(S_1,\ldots,S_N)$ belongs to the class ${\rm Nilp}_N(\Lambda)$. Indeed, if $v\in\mathcal{F}_N\setminus\Lambda$ and $w\in\Lambda$ then $\mathbf{S}^vw\neq 0$ implies $w=vw'$ with some $w'\in\Lambda$. But in this case (the set $\Lambda$ is admissible!) we get $w\in\mathcal{F}_N\setminus\Lambda$ which is impossible. Thus $\mathbf{S}^v=0$ for every $v\in\mathcal{F}_N\setminus\Lambda$. Since for every $w\in\Lambda$ we have $\mathbf{S}^ww=\emptyset\neq 0$, we obtain that $\mathbf{S}$ doesn't belong to the class ${\rm Nilp}_N(\tilde{\Lambda})$ for any admissible proper subset $\tilde{\Lambda}\subset\Lambda$. We can see also that $\mathbf{S}\in\mathcal{C}^N$: for  arbitrary
$x=\sum_{w\in\Lambda}x_ww\in\mathcal{H}_\Lambda$, with $x_w\in\mathbb{C}$ ($w\in\Lambda$), and $k\in\{ 1,\ldots,N\}$ we get
\begin{eqnarray*}
\| S_kx\|^2 &=& \left\|\sum_{w\in\Lambda}x_wS_kw\right\|^2=\left\|\sum_{w'\in\Lambda:\,g_kw'\in\Lambda}x_{g_kw'}w'\right\|^2\\
&=&
\sum_{w'\in\Lambda:\,g_kw'\in\Lambda}|x_{g_kw'}|^2\leq\sum_{w\in\Lambda}|x_w|^2=\| x\|^2,
\end{eqnarray*}
which means that $S_k$ are contractions.
\end{ex}
\begin{prop}\label{prop:vanish}
Let $\Lambda\subset\mathcal{F}_N$ be an admissible set. If a non-commutative polynomial $p(z)=\sum_{w\in\Lambda}p_wz^w\in\mathcal{L(U,Y)}\left\langle z_1,\ldots,z_N\right\rangle$, with some Hilbert spaces $\mathcal{U}$ and $\mathcal{Y}$, satisfies $p(\mathbf{T})=0$ for an arbitrary $N$-tuple of $\Lambda$-jointly nilpotent contractive $n\times n$ matrices, $n\in\mathbb{N}$,  then $p_w=0$ for all $w\in\Lambda$. Moreover, it suffices to take $n$ equal to $\# (\Lambda)$, the number of words in $\Lambda$.
\end{prop}
\begin{proof}
We have $p(\mathbf{S})=0$, where  $\mathbf{S}$ is the $N$-tuple of backward shifts from Example~\ref{ex:shifts}. Since $\mathbf{S}\in\mathcal{L}(\mathcal{H}_\Lambda)^N$, and $\dim \mathcal{H}_\Lambda =\# (\Lambda)$, one can consider 
$\mathbf{S}$ as an $N$-tuple of $\Lambda$-jointly nilpotent contractive $n\times n$ matrices, with $n=\# (\Lambda)$, as well as $\lambda\mathbf{S}:=(\lambda S_1,\ldots,\lambda S_N)$ for any $\lambda\in\mathbb{D}$. Therefore, a one-variable polynomial
$$r_\mathbf{S}(\lambda)=\sum_{k=0}^mr_{\mathbf{S},k}\lambda^k:=\sum_{k=0}^m\left(\sum_{w\in\Lambda:\,|w|=k}p_w\otimes\mathbf{S}^w\right)\lambda^k=\sum_{w\in\Lambda}p_w\otimes(\lambda\mathbf{S})^w=p(\lambda\mathbf{S}),$$
where $m=\max_{w\in\Lambda}|w|$, vanishes on $\mathbb{D}$, and hence vanishes identically. Thus $$r_{\mathbf{S},k}=\sum_{w\in\Lambda:\,|w|=k}p_w\otimes\mathbf{S}^w=0$$ for $k=0,\ldots,m$.
For a fixed $k$ and any $u\in\mathcal{U}$ and $v\in\Lambda:\ |v|=k$ (the word $v$ is identified with a basis vector in $\mathcal{H}_\Lambda$, or equivalently, with a standard basis vector in $\mathbb{C}^n$), we have:
$$
0=\sum_{w\in\Lambda:\,|w|=k}(p_w\otimes\mathbf{S}^w)(u\otimes v)=\sum_{w\in\Lambda:\,|w|=k}p_wu\otimes\mathbf{S}^wv=p_vu\otimes\emptyset.
$$
Since $\emptyset\neq 0$, we get $p_vu=0$. Since $k\in\{ 0,\ldots,m\}$, $v\in\Lambda$ and $u\in\mathcal{U}$ were chosen arbitrarily, the lemma follows.
\end{proof}
\begin{prop}\label{prop:jn}
Let $m\in\mathbb{N}$. An $N$-tuple $\mathbf{T}\in\mathcal{L(E)}^N$ belongs to the class ${\rm Nilp}_N(\Lambda_m)$ if and only if there exists a decomposition $\mathcal{E}=\mathcal{E}_1\oplus\cdots\oplus\mathcal{E}_{m+1}$ such that the operators $T_k,\ k=1,\ldots,N$, have strictly lower block-triangular form with respect to this decomposition:
$$T_k=\left[\begin{array}{cccc}
0      &   \ldots     &   \ldots     & 0\\
*      & \ddots & \ddots       & \vdots\\
\vdots & \ddots & \ddots & \vdots\\
*      & \ldots & *      & 0 
\end{array}
\right].$$
\end{prop} 
\begin{proof}
Clearly, any $N$-tuple $\mathbf{T}\in\mathcal{L(E)}^N$ of bounded linear operators on a Hilbert space $\mathcal{E}$ which have strictly lower block-triangular form with respect to some decomposition $\mathcal{E}=\mathcal{E}_1\oplus\cdots\oplus\mathcal{E}_{m+1}$ is $\Lambda_m$-jointly nilpotent.

Conversely, let $\mathbf{T}\in\mathcal{L(E)}^N$ be $\Lambda_m$-jointly nilpotent. Set
\begin{eqnarray*}
\mathcal{E}_1 &:=& \left(\bigvee_{w\in\mathcal{F}_N:\,|w|\geq 0}\mathbf{T}^w\mathcal{E}\right)\ominus\left(\bigvee_{w\in\mathcal{F}_N:\,|w|\geq 1}\mathbf{T}^w\mathcal{E}\right)\\
\cdots & \cdots & \cdots \\
\mathcal{E}_{m-1} &:=& \left(\bigvee_{w\in\mathcal{F}_N:\,|w|\geq m-2}\mathbf{T}^w\mathcal{E}\right)\ominus\left(\bigvee_{w\in\mathcal{F}_N:\,|w|\geq m-1}\mathbf{T}^w\mathcal{E}\right)\\
\mathcal{E}_m &:=& \left(\bigvee_{w\in\mathcal{F}_N:\,|w|\geq m-1}\mathbf{T}^w\mathcal{E}\right)\ominus\left(\bigvee_{w\in\mathcal{F}_N:\,|w|=m}\mathbf{T}^w\mathcal{E}\right)\\
\mathcal{E}_{m+1} &:=& \bigvee_{w\in\mathcal{F}_N:\,|w|= m}\mathbf{T}^w\mathcal{E},
\end{eqnarray*}
where $\bigvee_\nu \mathcal{X}_\nu$ denotes the closed linear span of the sets $\mathcal{X}_\nu$ ($\subset\mathcal{E}$).
Then $$\mathcal{E}=\bigoplus_{\nu=1}^{m+1}\mathcal{E}_\nu=\bigvee_{w\in\mathcal{F}_N:\,|w|\geq 0}\mathbf{T}^w\mathcal{E},$$ and $$T_k\mathcal{E}_{j}\subset\bigoplus_{\nu=j+1}^{m+1}\mathcal{E}_\nu=\bigvee_{w\in\mathcal{F}_N:\,|w|\geq j}\mathbf{T}^w\mathcal{E},\quad T_k\mathcal{E}_{m+1}=\{ 0\},\quad k=1,\ldots,N,\ j=1,\ldots,m,$$
which means that $T_k,\ k=1,\ldots,N,$ have strictly lower block-triangular form with respect to the decomposition $\mathcal{E}=\mathcal{E}_1\oplus\cdots\oplus\mathcal{E}_{m+1}$.
\end{proof}
\begin{rem}\label{rem:nm}
The special case of Proposition~\ref{prop:jn} where $\dim\mathcal{E}<\infty$ can be formulated as follows: an $N$-tuple of matrices $\mathbf{T}=(T_1,\ldots,T_N)\in (\mathbf{C}^{n\times n})^N$ is jointly nilpotent, with rank of joint nilpotency at most $m+1$, if and only if $\mathbf{T}$ is unitary similar to an $N$-tuple $\tilde{\mathbf{T}}=(\tilde{T}_1,\ldots,\tilde{T}_N)$ of strictly lower block-triangular $(m+1)\times (m+1)$ matrices,  with not necessarily square blocks. (Here unitary similarity means that there exists a unitary $n\times n$ matrix $U$ such that $T_k=U^{-1}\tilde{T}_kU,\ k=1,\ldots,N$.) This statement is a bit stronger than one in \cite{KV} where only similarity of an $N$-tuple of jointly nilpotent matrices to some $N$-tuple of strictly triangular matrices was mentioned.  
\end{rem}
\begin{lem}\label{lem:dil}
Let $\Lambda\subset\mathcal{F}_N$ be an admissible set, $\mathbf{U}\in\mathcal{U}^N\cap\mathcal{L(K)}^N$, and let $W\in\mathcal{L(Y,K)}$ be an isometry, with Hilbert spaces $\mathcal{Y}$ and $\mathcal{K}$, such that
\begin{equation}\label{cond}
W^*\mathbf{U}^wW=0 \quad {\rm for\ } w\in\mathcal{F}_N\setminus\Lambda.
\end{equation}
Then there exist a Hilbert space $\mathcal{E}$ and an $N$-tuple $\mathbf{T}\in\mathcal{C}^N\cap\mathcal{L(E)}^N$ of $\Lambda$-jointly nilpotent operators such that 
$$
\mathcal{K}\supset\mathcal{E}\supset W\mathcal{Y}
$$
and $\mathbf{U}$ is a unitary dilation of $\mathbf{T}$:
$$\mathbf{T}^w=P_\mathcal{E}\mathbf{U}^w\big|_\mathcal{E},\qquad w\in\mathcal{F}_N.$$
In particular,
\begin{equation}\label{dil}
W^*\mathbf{U}^wW=\tilde{W}^*\mathbf{T}^w\tilde{W},\quad w\in\mathcal{F}_N,
\end{equation}
where $\tilde{W}=P_\mathcal{E}W\in\mathcal{L(Y,E)}$ is an isometry.
If the space $\mathcal{Y}$ is finite-dimensional then one can choose $\mathcal{E}$ finite-dimensional, too.
\end{lem}
\begin{proof}
Define the following subspaces in $\mathcal{K}$:
\begin{eqnarray*}
\mathcal{E}_0 &:=& \bigvee_{w\in\mathcal{F}_N\setminus\Lambda}\mathbf{U}^wW\mathcal{Y},\\
\mathcal{E}   &:=& \left(\bigvee_{w\in\mathcal{F}_N}\mathbf{U}^wW\mathcal{Y}\right)\ominus\left(\bigvee_{w\in\mathcal{F}_N\setminus\Lambda}\mathbf{U}^wW\mathcal{Y}\right),
\end{eqnarray*}
and define the operators $$T_k:=P_\mathcal{E}U_k\big|_\mathcal{E},\quad k=1,\ldots,N.$$
Clearly, $\mathbf{T}=(T_1,\ldots,T_N)\in\mathcal{C}^N$. 
Since both $\mathcal{E}_0$ and $\mathcal{E}_0\oplus\mathcal{E}=\bigvee_{w\in\mathcal{F}_N}\mathbf{U}^wW\mathcal{Y}$ are invariant subspaces for every $U_k,\ k=1,\ldots,N$,  the space $\mathcal{E}$ is an orthogonal difference of two invariant subspaces for these unitary operators, i.e., a \emph{semi-invariant subspace}. Thus, by the Sarason lemma \cite[Lemma~0]{Sar2} $\mathbf{U}$ is a unitary dilation of $\mathbf{T}$. 
Since for every $w\in\mathcal{F}_N\setminus\Lambda$ one has $\mathbf{U}^w\mathcal{E}\subset\mathcal{E}_0$, we get $\mathbf{T}^w=P_\mathcal{E}\mathbf{U}^w\big|_\mathcal{E}=0$, i.e., $\mathbf{T}\in {\rm Nilp}_N(\Lambda)$.
Since by \eqref{cond} $W^*\mathcal{E}_0=\{ 0\}$,  and $W\mathcal{Y}\subset\mathcal{E}_0\oplus\mathcal{E}=\bigvee_{w\in\mathcal{F}_N}\mathbf{U}^wW\mathcal{Y}$, we 
get $W\mathcal{Y}\subset\mathcal{E}$, as desired. Thus, \eqref{dil} holds true as well, with an isometry $\tilde{W}=P_\mathcal{E}W\in\mathcal{L(Y,E)}$. 

In the case where $\dim\mathcal{Y}<\infty$, we may write
$$ \mathcal{E}_0\oplus\mathcal{E}=\left(\bigvee_{w\in\mathcal{F}_N\setminus\Lambda}\mathbf{U}^wW\mathcal{Y}\right)\oplus P_\mathcal{E}\left(\bigvee_{w\in\Lambda}\mathbf{U}^wW\mathcal{Y}\right)=\mathcal{E}_0\oplus P_\mathcal{E}\left(\bigvee_{w\in\Lambda}\mathbf{U}^wW\mathcal{Y}\right),$$ 
and since the set $\Lambda$ is finite, both $\bigvee_{w\in\Lambda}\mathbf{U}^wW\mathcal{Y}$ and $\mathcal{E}=P_\mathcal{E}\left(\bigvee_{w\in\Lambda}\mathbf{U}^wW\mathcal{Y}\right)$ are finite-dimensional subspaces.
\end{proof}
\begin{thm}\label{thm:main}
Problem~\ref{prob:c-gen} has a solution if and only if the polynomial 
\begin{equation}\label{poly}
p(z):=\frac{c_\emptyset}{2}+\sum_{w\in\Lambda\setminus\{\emptyset\} }c_wz^w
\end{equation}
satisfies $\Re p(\mathbf{T})\geq 0$ for every $N$-tuple $\mathbf{T}$ of $\Lambda$-jointly nilpotent contractive operators. Moreover, for the solvability of Problem~\ref{prob:c-gen} it is enough to assume that $\Re p(\mathbf{T})\geq 0$ holds for every $N$-tuple $\mathbf{T}$ of $\Lambda$-jointly nilpotent contractive $n\times n$ matrices, for all $n\in\mathbb{N}$.
\end{thm}
\begin{proof}
If Problem~\ref{prob:c-gen} has a solution $f\in\mathcal{HA}_N^{\rm nc}(\mathcal{Y})$ then  for any $\mathbf{C}\in\mathcal{D}^N\cap\mathcal{L(E)}^N$, with a Hilbert space $\mathcal{E}$, the series $$f(\mathbf{C}):=\sum_{w\in\mathcal{F}_N}f_w\otimes\mathbf{C}^w$$
converges in the operator norm, and $\Re f(\mathbf{C})\geq 0$. If $\mathbf{T}\in\mathcal{C}^N\cap\mathcal{L(E)}^N$ is an $N$-tuple of $\Lambda$-jointly nilpotent operators then so is $r\mathbf{T}=(rT_1,\ldots,rT_N)\in\mathcal{D}^N\cap\mathcal{L(E)}^N$ for every $r:\ 0<r<1$. Therefore, $$\Re p(r\mathbf{T})=\Re f(r\mathbf{T})=\Re\left(\frac{c_\emptyset\otimes I_\mathcal{E}}{2}+\sum_{w\in\Lambda\setminus\{ \emptyset\}}c_w\otimes(r\mathbf{T})^w\right)\geq 0.$$
By letting $r\uparrow 1$, we obtain $\Re p(\mathbf{T})\geq 0$.

For the converse direction, let us consider first the case where $c_\emptyset=I_\mathcal{Y}$. Let $\Re p(\mathbf{T})\geq 0$ hold for every $N$-tuple $\mathbf{T}$ of $\Lambda$-jointly nilpotent contractive $n\times n$ matrices, for all $n\in\mathbb{N}$. Let $\mathcal{A}_{\mathcal{G}_N}$ be the $C^*$-algebra obtained as  the norm completion of the quotient of unital $*$-algebra $\mathcal{A}_N$ (which has been introduced in the proof of Lemma~\ref{lem:main} above) with the seminorm
$$\| q\| :=\sup_{\mathbf{G}\in\mathcal{G}_N}\| q(\mathbf{G})\|=\sup_{\mathbf{G}\in\mathcal{G}^N}\| q(G_1,\ldots,G_N,G_1^*,\ldots,G_N^*)\|,$$
by the two-sided ideal of elements of zero seminorm. 

Let us show that the restriction of the quotient map above to the subspace $\mathcal{B}_N\subset\mathcal{A}_N$ of polynomials of the form \eqref{q}
 is injective, i.e., that if $q\in\mathcal{B}_N$ is non-zero then the corresponding coset $[q]\in\mathcal{A}_{\mathcal{G}_N}$ is non-zero. Indeed, if $[q]=[0]$ then $q(\mathbf{G})=0$ for every $\mathbf{G}\in\mathcal{G}_N\cap\mathcal{L}(\mathcal{H})^N$, with a Hilbert space $\mathcal{H}$. Define $\tilde{q}(\hat{z}):=1+q(\hat{z})$. Then $\tilde{q}(\mathbf{G})=I_{\mathcal{H}}>0$ for every $\mathbf{G}\in\mathcal{G}_N\cap\mathcal{L}(\mathcal{H})^N$. In particular, $\tilde{q}(\mathbf{G}_{\zeta})=1$ for $\mathbf{G}_{\zeta}:=(\zeta,0,\ldots,0)\in\mathcal{G}_N\cap\mathbb{C}^N,\ \zeta\in\mathbb{T}$. Therefore,
$$\tilde{q}_\emptyset=\int_{\mathbb{T}}\tilde{q}(\mathbf{G}_\zeta)\,d\zeta =1,$$
and $q_\emptyset=0$.
 By Part~II of Lemma~\ref{lem:main}, there exist a Hilbert space $\mathcal{K}$, an $N$-tuple of operators $\mathbf{U}\in\mathcal{U}^N\cap\mathcal{L}(\mathcal{K})^N$, and an isometry $W\in\mathcal{L}(\mathbb{C},\mathcal{K})$ such that 
	\begin{eqnarray*}
	\tilde{q}_w=\tilde{q}_{w^*}^* &=&  W^*\mathbf{U}^wW,\qquad w\in\mathcal{F}_N:\,|w|\leq m,\\
	0 &=& W^*\mathbf{U}^wW,\qquad w\in \mathcal{F}_N:\, |w|>m.
\end{eqnarray*}
Then $$q_w=\tilde{q}_w=\tilde{q}_{w^*}^*=q_{w^*}^*=W^*\mathbf{U}^w\tilde{W},\qquad w\in\mathcal{F}_N:\,0<|w|\leq m.$$
For an arbitrary $\mathbf{G}\in\mathcal{G}_N\cap\mathcal{L}(\mathcal{H})^N$ the one-variable trigonometric polynomial $$t_\mathbf{G}(\zeta):=q(\zeta\mathbf{G})=\sum_{k=1}^m\left(\sum_{w\in\mathcal{F}_N:\,|w|=k}q_w\mathbf{G}^w\right)\zeta^k+\sum_{k=1}^m\left(\sum_{w\in\mathcal{F}_N^*:\,|w|=k}q_w\mathbf{G}^{*w}\right)\bar{\zeta}^k$$
is identically zero, which implies in particular
$$\sum_{w\in\mathcal{F}_N:\,|w|=k}q_w\mathbf{G}^w=0,\qquad  k=1,\ldots,m.$$
For arbitrary $n\in\mathbb{N}$ and  $\tilde{\mathbf{U}}\in\mathcal{U}^N\cap {(\mathbb{C}^{n\times n})}^N$, by virtue of Proposition~\ref{prop:schur} we have $\mathbf{G}\stackrel{\circ}{\otimes }\tilde{\mathbf{U}}\in\mathcal{G}_N$, therefore
$$\sum_{w\in\mathcal{F}_N:\,|w|=k}q_w(\mathbf{G}\stackrel{\circ}{\otimes}\tilde{\mathbf{U}})^w=\sum_{w\in\mathcal{F}_N:\,|w|=k}q_w\mathbf{G}^w\otimes\tilde{\mathbf{U}}^w=0,\qquad  k=1,\ldots,m.$$
Since $\mathcal{U}^N\cap {(\mathbb{C}^{n\times n})}^N$ is a uniqueness set for holomorphic functions of matrix entries (see e.g., \cite{Sh}), the non-commutative polynomial $\sum_{w\in\mathcal{F}_N:\,|w|=k}q_w\mathbf{G}^wz^w$ vanishes on the all of ${(\mathbb{C}^{n\times n})}^N$, for every $k\in\{ 1,\ldots,m\}$ and $n\in\mathbb{N}$:
$$\sum_{w\in\mathcal{F}_N:\,|w|=k}q_w\mathbf{G}^w\otimes \mathbf{Z}^w=0,\qquad \mathbf{Z}\in {(\mathbb{C}^{n\times n})}^N.$$ Thus, by the Amitsur--Levitzki theorem (see \cite[pp. 22--23]{Ro}), $$q_w\mathbf{G}^w=0,\qquad w\in\mathcal{F}_N:\ 0<|w|\leq m.$$ For every $w\in\mathcal{F}_N:\ 0<|w|\leq m$ one can find $\mathbf{G}\in\mathcal{G}_N$ such that 
$\mathbf{G}^w\neq 0$. Indeed, the non-commutative polynomial $\psi(z)=1+z^w$ belongs to the class $\mathcal{HA}_N^{{\rm nc}, 1}=\mathcal{HA}_N^{{\rm nc},1}(\mathbb{C})$, thus by Corollary~\ref{cor:coef}, there exist a Hilbert space $\mathcal{H}$, an $N$-tuple of operators  $\mathbf{G}\in\mathcal{G}_N\cap\mathcal{L}(\mathcal{H})^N$, and an isometry $V\in\mathcal{L}(\mathbb{C}, \mathcal{H})$ such that $\psi_w=1=2V^*\mathbf{G}^wV$, which implies $\mathbf{G}^w\neq 0$. Therefore, $q_w=0$ for all $w\in\mathcal{F}_N:\ 0<|w|\leq m$. Since we have shown already that $q_\emptyset=0$, and $q_w=q_{w^*}^*$ for $w\in\mathcal{F}_N:\ 0<|w|\leq m$, we get $q=0$.

Denote by $\mathcal{B}_\Lambda\subset\mathcal{B}_N$ the finite-dimensional subspace of polynomials of the form
\begin{equation}\label{ql}
q(\hat{z})=q_\emptyset+\sum_{w\in\Lambda\setminus\{\emptyset\} }q_wz^w+\sum_{w\in\Lambda^*\setminus\{\emptyset\} }q_wz^{*w},
\end{equation}
and let $\mathcal{B}_{\Lambda,\mathcal{G}_N}$ be the image of the subspace $\mathcal{B}_\Lambda$ under the quotient map above. This subspace of the $C^*$-algebra $\mathcal{A}_{\mathcal{G}_N}$ is selfadjoint, i.e., $\mathcal{B}_{\Lambda,\mathcal{G}_N}^*=\mathcal{B}_{\Lambda,\mathcal{G}_N}.$ Define the linear map $\varphi:\ \mathcal{B}_{\Lambda,\mathcal{G}_N}\to\mathcal{L(Y)}$ by $\varphi([1])=I_\mathcal{Y}$, $\varphi([z^w])=c_w$ for $w\in\Lambda\setminus\{ \emptyset\}$, and $\varphi([z^{*w}])=c_{w^*}^*$ for $w\in\Lambda^*\setminus\{ \emptyset\}$. By the previous paragraph together with the above mentioned Amitsur--Levitzki theorem, this linear map is correctly defined. Let us show that $\varphi$ is completely positive. Let $n\in\mathbb{N}$, and let $[q]\in\mathbb{C}^{n\times n}\otimes\mathcal{B}_{\Lambda,\mathcal{G}_N}$ be a positive element of the $C^*$-algebra $\mathbb{C}^{n\times n}\otimes\mathcal{A}_{\mathcal{G}_N}$, i.e.,
$[q]=[h]^*[h]$ with some $[h]\in\mathbb{C}^{n\times n}\otimes\mathcal{A}_{\mathcal{G}_N}$. One can think of $[q]$ as of the $n\times n$ matrix $([q]_{ij})_{i,j=1,\ldots,n}$ whose entries $[q]_{ij}=[q_{ij}]\in\mathcal{B}_{\Lambda,\mathcal{G}_N}$ and $q_{ij}\in\mathcal{B}_\Lambda$, and thus $q\in\mathbb{C}^{n\times n}\otimes\mathcal{B}_\Lambda$ is a polynomial of the form \eqref{ql} with the coefficients from $\mathbb{C}^{n\times n}$. Let us observe that by virtue of the definition of the $C^*$-algebra $\mathcal{A}_{\mathcal{G}_N}$, for an arbitrary $[x]\in\mathcal{A}_{\mathcal{G}_N}$ its values on $\mathcal{G}_N$ are well defined. In particular, if $x\in\mathcal{B}_\Lambda$ then $[x](\mathbf{G})=x(\mathbf{G})$ for any $\mathbf{G}\in\mathcal{G}_N$. Therefore, for an arbitrary $[x]=([x]_{ij})_{i,j=1,\ldots,n}=([x_{ij}])_{i,j=1,\ldots,n}\in\mathbb{C}^{n\times n}\otimes\mathcal{A}_{\mathcal{G}_N}$ one defines correctly $[x](\mathbf{G}):=([x_{ij}](\mathbf{G}))_{i,j=1,\ldots,n},\ \mathbf{G}\in\mathcal{G}_N$. In particular, if $x=(x_{ij})_{i,j=1,\ldots,n}\in\mathbb{C}^{n\times n}\otimes\mathcal{B}_\Lambda$ then $[x](\mathbf{G})=x(\mathbf{G})=(x_{ij}(\mathbf{G}))_{i,j=1,\ldots,n}$ for any $\mathbf{G}\in\mathcal{G}_N$. Since $q(\mathbf{G})=[q](\mathbf{G})=[h](\mathbf{G})^*[h](\mathbf{G})$ is positive semidefinite for every  $\mathbf{G}\in\mathcal{G}_N$, it follows that the polynomial $q$ (with the coefficients from $\mathbb{C}^{n\times n}$) is positive semidefinite on $\mathcal{G}_N$. In particular, $q(\mathbf{G}_{\zeta})\geq 0$ for $\mathbf{G}_{\zeta}:=(\zeta,0,\ldots,0)\in\mathcal{G}_N\cap\mathbb{C}^N,\ \zeta\in\mathbb{T}$. Therefore,
$$q_\emptyset=\int_{\mathbb{T}}q(\mathbf{G}_\zeta)\,d\zeta\geq 0.$$
  If $q_\emptyset=I_n$, then by Part II of Lemma~\ref{lem:main} there exist a Hilbert space $\mathcal{K}$, an $N$-tuple of operators $\mathbf{U}\in\mathcal{U}^N\cap\mathcal{L}(\mathcal{K})^N$, and an isometry $W\in\mathcal{L}(\mathbb{C}^n,\mathcal{K})$ such that 
\begin{eqnarray*}
q_w=q_{w^*}^* &=& W^*\mathbf{U}^wW,\qquad w\in\Lambda,\\
0   &=& W^*\mathbf{U}^wW,\qquad w\in\mathcal{F}_N\setminus\Lambda.
\end{eqnarray*}
Then we have
\allowdisplaybreaks
\begin{eqnarray*}
\varphi_n([q]) &=& ({\rm id}_n\otimes \varphi)\left(I_n\otimes [1]+\sum_{w\in\Lambda\setminus\{\emptyset\} }q_w\otimes [z^w]+ \sum_{w\in\Lambda^*\setminus\{\emptyset\} }q_w\otimes [z^{*w}] \right)\\ 
&=& I_{\mathbb{C}^n\otimes \mathcal{Y}}+\sum_{w\in\Lambda\setminus\{\emptyset\} }W^*\mathbf{U}^wW\otimes c_w
+ \sum_{w\in\Lambda^*\setminus\{\emptyset\} }W^*\mathbf{U}^{*w}W\otimes c_{w^*}^*\\
&=& (W^*\otimes I_\mathcal{Y})2\Re p^{\rm l}(\mathbf{U})(W\otimes I_\mathcal{Y})
\end{eqnarray*}
By Lemma~\ref{lem:dil}, there exist a finite-dimensional Hilbert space $\mathcal{E}$ and an $N$-tuple $\mathbf{T}\in\mathcal{C}^N\cap\mathcal{L(E)}^N$ of $\Lambda$-jointly nilpotent operators such that 
\eqref{dil} holds
with an isometry $\tilde{W}\in\mathcal{L}(\mathbb{C}^n,\mathcal{E})$. Thus,
$$\varphi_n([q])=(\tilde{W}^*\otimes I_\mathcal{Y})2\Re p^{\rm l}(\mathbf{T})(\tilde{W}\otimes I_\mathcal{Y})$$
is a positive semidefinite operator by the assumption that $\Re p(\mathbf{T})\geq 0$ or, equivalently, $\Re 
p^{\rm l}(\mathbf{T})\geq 0$.
In the case where $q_\emptyset >0$ we can define $\tilde{q}(\hat{z}):=q_\emptyset^{-1/2}q(\hat{z})q_\emptyset^{-1/2}$. Since $\varphi_n([\tilde{q}])\geq 0$, we get $$\varphi_n([q])=(q_\emptyset^{1/2}\otimes I_\mathcal{Y})\varphi_n([\tilde{q}])(q_\emptyset^{1/2}\otimes I_\mathcal{Y})\geq 0.$$
In the case where the matrix $q_\emptyset$ is degenerate we set $q_\epsilon(\hat{z}) :=\epsilon I_n+q(\hat{z})$ for  $\epsilon >0$. Then $q_\epsilon$ is positive definite on $\mathcal{G}_N$ and ${(q_\epsilon)}_\emptyset=\epsilon I_n+q_\emptyset >0$. Since $\varphi_n([q_\epsilon])\geq 0$, we get
 $$
\varphi_n([q]) = \lim_{\epsilon\downarrow 0}\varphi_n([q_\epsilon ])\geq 0.
$$
Finally, we have obtained that $\varphi: \mathcal{B}_{\Lambda,\mathcal{G}_N}\rightarrow \mathcal{L(Y)}$ is completely positive.

 Since we have $\varphi ([1])=I_\mathcal{Y}$, by the Arveson extension theorem \cite{Arv1} there exists a completely positive map $\widetilde{\varphi}:\mathcal{A}_{\mathcal{G}_N}\rightarrow \mathcal{L(Y)}$ which extends $\varphi$. By the Stinespring theorem \cite{St}, there exists a $*$-representation $\pi$ of $\mathcal{A}_{\mathcal{G}_N}$ in some Hilbert space $\mathcal{H}$ and an isometry $V\in\mathcal{L(Y,H)}$ such that
$$\widetilde{\varphi}(a)=V^*\pi(a)V,\qquad a\in\mathcal{A}_{\mathcal{G}_N}.$$
In particular, we get
\begin{equation}
c_w = \varphi([z^w])=\widetilde{\varphi}([z^w])=V^*\pi([z^w])V=V^*\mathbf{G}^wV,\quad w\in\Lambda,\label{II1''}\\
\end{equation}
where we set $\mathbf{G}:=(\pi([z_1]),\ldots,\pi([z_N]))$. We have $\mathbf{G}\in\mathcal{G}_N$. Indeed, since by Proposition~\ref{prop:gn} we have $$\left\| \left[1-\sum_{k=1}^Nz_k^*z_k\right]\right\|_{\mathcal{A}_{\mathcal{G}_N}}=\left\| \left[1-\sum_{k=1}^Nz_kz_k^*\right]\right\|_{\mathcal{A}_{\mathcal{G}_N}}=0,$$
and $$\|[z_k^*z_j]\|_{\mathcal{A}_{\mathcal{G}_N}}=\|[z_kz_j^*]\|_{\mathcal{A}_{\mathcal{G}_N}}=0,\quad k\neq j,$$
 we get $[\sum_{k=1}^Nz_k^*z_k]=[\sum_{k=1}^Nz_kz_k^*]=[1]$, and $[z_k^*z_j]=[z_kz_j^*]=[0]$ for $k\neq j$. Hence
\begin{eqnarray*}
	\sum_{k=1}^NG_k^*G_k &=& \sum_{k=1}^N\pi([z_k])^*\pi([z_k])=\pi\left(\left[ \sum_{k=1}^Nz_k^*z_k\right]\right)=\pi([1])=I_\mathcal{H},\\
	G_k^*G_j &=& \pi([z_k])^*\pi([z_j])=\pi([z_k^*z_j])=\pi([0])=0,\qquad k\neq j,\\
	\sum_{k=1}^NG_kG_k^* &=& \sum_{k=1}^N\pi([z_k])\pi([z_k])^*=\pi\left(\left[ \sum_{k=1}^Nz_kz_k^*\right]\right)=\pi([1])=I_\mathcal{H},\\
	G_kG_j^* &=& \pi([z_k])\pi([z_j])^*=\pi([z_kz_j^*])=\pi([0])=0,\qquad k\neq j,
\end{eqnarray*} 
which means that $\mathbf{G}\in\mathcal{G}_N$, according to Proposition~\ref{prop:gn}.
	Finally, since \eqref{II1''} coincides with \eqref{mom-nc}, from Theorem~\ref{thm:g} we obtain that  Problem~\ref{prob:c} has a solution.

Consider now the case where $c_\emptyset >0$. Set $\widetilde{c_w}:=c_\emptyset^{-1/2}c_wc_\emptyset^{-1/2},\ w\in\Lambda$, and $\widetilde{p}(z):=c_\emptyset^{-1/2}p(z)c_\emptyset^{-1/2}$, where $p(z)$ is given by \eqref{poly}. Clearly, $\widetilde{c_\emptyset }=I_\mathcal{Y}$. Since Problem~\ref{prob:c} for the data $\widetilde{c_w},\ w\in\Lambda$, is solvable if and only if $\Re\widetilde{p}(\mathbf{T})\geq 0$ for every $N$-tuple $\mathbf{T}$ of contractive $\Lambda$-jointly nilpotent square matrices of same size, and since $\Re\widetilde{p}(\mathbf{T})\geq 0\Leftrightarrow\Re p(\mathbf{T})\geq 0$ and $f\in\mathcal{HA}_N^{\rm nc}(\mathcal{Y})\Leftrightarrow c_\emptyset^{-1/2}fc_\emptyset^{-1/2}\in\mathcal{HA}_N^{\rm nc}(\mathcal{Y})$, Problem~\ref{prob:c-gen} for the data $c_\emptyset>0,\ c_w\ (w\in\Lambda\setminus\{\emptyset\} )$ is solvable if and only if $\Re p(\mathbf{T})\geq 0$ for every $N$-tuple $\mathbf{T}$ of contractive $\Lambda$-jointly nilpotent square matrices of same size.

Consider now the general case $c_\emptyset\geq 0$. Suppose that $\Re p(\mathbf{T})\geq 0$ for every $N$-tuple $\mathbf{T}$ of contractive $\Lambda$-jointly nilpotent $n\times n$ matrices, for all $n\in\mathbb{N}$. Then for any such $\mathbf{T}$ the polynomial $$f_\mathbf{T}(\lambda):=p(\lambda\mathbf{T})=\frac{c_\emptyset\otimes I_n}{2}+\sum_{k=1}^m \left(\sum_{w\in\Lambda:\,|w|=k}c_w\otimes\mathbf{T}^w\right)\lambda^k,\quad \lambda\in\mathbb{D},$$
where $m=\max_{w\in\Lambda}|w|$,
belongs to the Herglotz class $\mathcal{H}_1(\mathcal{Y}\otimes\mathbb{C}^{n})$. Since its coefficients are   $\frac{c_0(\mathbf{T})}{2}:=(f_\mathbf{T})_0=\frac{c_\emptyset\otimes I_n}{2}$, $c_k(\mathbf{T}):=(f_\mathbf{T})_k=\sum_{w\in\Lambda:\,|w|=k}c_w\otimes\mathbf{T}^w,\ k=1,\ldots,m,$  from the Carath\'{e}odory--Toeplitz criterion of solvability of the one-variable Carath\'{e}odory problem with data $c=\{ c_k(\mathbf{T})\}_{k=0,\ldots,m}$ we obtain $T_c\geq 0$, where the operator block matrix $T_c$ is defined by \eqref{t}.
The condition $T_c\geq 0$ implies for $k=1,\ldots,m$ the following inequalities:
\begin{equation}\label{ineq}
|\left\langle c_k(\mathbf{T})x,y\right\rangle |^2\leq\left\langle (c_\emptyset\otimes I_n) x,x\right\rangle\left\langle (c_\emptyset\otimes I_n) y,y\right\rangle,\quad x,y\in\mathcal{Y}\otimes\mathbb{C}^n,
\end{equation}
which yield  $\ker c_\emptyset\otimes\mathbb{C}^n\subset \ker c_k(\mathbf{T})$, and $\ker c_\emptyset\otimes\mathbb{C}^n\subset\ker c_k(\mathbf{T})^*$. Let $x\in\ker c_\emptyset$ and $k\in\{ 1,\ldots,m\}$. Then the non-commutative polynomial $\sum_{w\in\Lambda:\,|w|=k}(c_wx)z^w$ with coefficients in $\mathcal{Y}\cong\mathcal{L}(\mathbb{C},\mathcal{Y})$ vanishes on  $N$-tuples of contractive  $\Lambda$-jointly nilpotent $n\times n$ matrices, for every $n\in\mathbb{N}$. By Proposition~\ref{prop:vanish}, $c_wx=0$ for all $w\in\Lambda:\ |w|=k$.
 Therefore, for every $w\in\Lambda$ we have $\ker c_\emptyset\subset\ker c_w$. Analogously,  $\ker c_\emptyset\subset\ker c_w^*$ for every $w\in\Lambda$. We obtain that our data of Problem~\ref{prob:c-gen} have the following operator block matrix form with respect to the decomposition $\mathcal{Y}=\ker c_\emptyset\otimes {\rm ran}\,c_\emptyset$:
 $$c_w=\left[\begin{array}{cc}
0 & 0\\
0 & c_w^{(22)}\end{array}\right],\quad w\in\Lambda.$$
 Since $c_\emptyset^{(22)}>0$, and the polynomial $$p^{(22)}(z):=\frac{c_\emptyset^{(22)}}{2}+\sum_{w\in\Lambda\setminus\{\emptyset\} }c_w^{(22)}z^w$$
 satisfies the condition that $\Re p^{(22)}(\mathbf{T})\geq 0$ for every $N$-tuple $\mathbf{T}$ of contractive $\Lambda$-jointly nilpotent square matrices of same size, by the result of the previous paragraph, Problem~\ref{prob:c-gen} for the data $c_w^{(22)},\ w\in\Lambda$, has a solution $f^{(22)}\in\mathcal{HA}_N^{\rm nc}({\rm ran}\,c_\emptyset)$. Then Problem~\ref{prob:c-gen} for the data $c_w,\ w\in\Lambda$, has a solution
 $$f=\left[\begin{array}{cc}
0 & 0\\
0 & f^{(22)}\end{array}\right]\in\mathcal{HA}_N^{\rm nc}(\mathcal{Y}).$$
\end{proof}

Let us remark that the condition that $\Re p(\mathbf{T})\geq 0$ for every $N$-tuple of $\tilde{\Lambda}$-jointly nilpotent contractive $n\times n$ matrices, for all $n\in\mathbb{N}$, where $\tilde{\Lambda}\supset\Lambda$ is a wider admissible set, is sufficient for the solvability of Problem~\ref{prob:c-gen}. For instance, one might find convenient to test this condition for the set $\tilde{\Lambda}=\Lambda_m$, with $m=\max_{w\in\Lambda}|w|$, and use the structure of $\Lambda_m$-jointly nilpotent matrices described in Remark~\ref{rem:nm}. However, one should remember that in general this condition is not necessary for the solvability of Problem~\ref{prob:c-gen}.
\begin{ex}\label{ex:1}
Let $\Lambda=\{\emptyset,g_1,g_2,g_1g_2,g_2g_1\}\in\mathcal{F}_2$, and the scalar data of the Carath\'{e}odory problem are
$c_\emptyset=1,\ c_{g_1}=c_{g_2}=\frac{1}{2},\ c_{g_1g_2}=c_{g_2g_1}=\frac{1}{4}$. Then the formal power series
$$\frac{1}{2}\left(1+\frac{z_1+z_2}{2}\right)\left(1-\frac{z_1+z_2}{2}\right)^{-1}=\frac{1}{2}+\sum_{j=1}^\infty\left(\frac{z_1+z_2}{2}\right)^j\in\mathcal{HA}_2^{\rm nc}$$
is a solution to this problem. By Theorem~\ref{thm:main}, for the polynomial $$p(z):=\frac{1}{2}+\frac{z_1+z_2}{2}+\frac{z_1z_2+z_2z_1}{4}$$ and  for every pair of $n\times n$ matrices $\mathbf{T}=(T_1,T_2)\in\mathcal{C}^2\cap {\rm Nilp}_2(\Lambda),\ n\in\mathbb{N}$, one has $\Re p(\mathbf{T})\geq 0$.
Let $\tilde{\Lambda}:=\{\emptyset,g_1,g_2,g_1g_2,g_2g_1,g_1^2\}\supset\Lambda$, and $\mathbf{T}=(T_1,T_2)\in (\mathbb{C}^{3\times 3})^2$ be given by
$$T_1:=\left[\begin{array}{ccc}
0 & 1 & 0\\
0 & 0 & 1\\
0 & 0 & 0\end{array}\right],\quad T_2:=\left[\begin{array}{ccc}
0 & 1 & 0\\
0 & 0 & 0\\
0 & 0 & 0\end{array}\right].$$
It is easy to see that $\mathbf{T}\in\mathcal{C}^2\cap {\rm Nilp}_2(\tilde{\Lambda})$, and $\mathbf{T}\notin {\rm Nilp}_2(\Lambda)$ since $T_1^2\neq 0$. Since $$\det (2\Re p(\mathbf{T}))=\left|\begin{array}{ccc}
1 & 1 & 1/4\\
1 & 1 & 1/2\\
1/4 & 1/2 & 1\end{array}\right| =-\frac{1}{16}<0,$$
the condition $\Re p(\mathbf{T})\geq 0$ is not fulfilled. The same is true for the (admissible) set $\tilde{\Lambda}:=\{\emptyset,g_1,g_2,g_1g_2,g_2g_1,g_2^2\}\supset\Lambda$, where we take the previous example of $\mathbf{T}=(T_1,T_2)$ and interchange $T_1\leftrightarrow T_2$. Clearly, the condition $\Re p(\mathbf{T})\geq 0$ can not be fulfilled for all pairs of $n\times n$ matrices $\mathbf{T}\in\mathcal{C}^2\cap {\rm Nilp}_2(\Lambda_2),\ n\in\mathbb{N}$.
\end{ex}
The following example shows that sometimes the above mentioned condition for a wider set $\tilde{\Lambda}\supset \Lambda$
is necessary for the solvability of Problem~\ref{prob:c-gen}.
\begin{ex}\label{ex:2}
Let $\Lambda=\{\emptyset,g_1,g_1^2,\ldots,g_1^m\}\subset\mathcal{F}_N$ and $c_\emptyset\geq 0,c_{g_1},c_{g_1^2},\ldots,c_{g_1^m}\in\mathcal{L(Y)}$, with some $m\in\mathbb{N}$ and some Hilbert space $\mathcal{Y}$. Then the class ${\rm Nilp}_N(\Lambda)$ consists of $N$-tuples of operators of the form $\mathbf{T}=(T_1,0,\ldots,0)$, where $T_1$ is a nilpotent operator. The condition that $\Re p(\mathbf{T})=\Re (\frac{c_\emptyset\otimes I}{2}+\sum_{j=1}^mc_{g_1^j}\otimes T_1^j)\geq 0$ for every nilpotent contractive square matrix $T_1$ is necessary and sufficient for the solvability of Problem~\ref{prob:c-gen} for these data (and equivalent to the Carath\'{e}odory--Toeplitz criterion for the solvability of a one-variable Carath\'{e}odory problem, see Section~\ref{sec:intro}). Then for every admissible set $\tilde{\Lambda}\supset\Lambda$ such that $g_1^{m+1}\notin\tilde{\Lambda}$ the condition $\Re p(\mathbf{T})\geq 0$ is fulfilled for every $N$-tuple of matrices $\mathbf{T}\in\mathcal{C}^N\cap {\rm Nilp}_N(\tilde{\Lambda})$. In particular it is fulfilled for $\tilde{\Lambda}=\Lambda_m$.
\end{ex}
Thus, natural open questions are the following.
For which admissible sets $\Lambda\subset\mathcal{F}_N$ and data $c_w,\ w\in\Lambda$, the condition that $\Re p(\mathbf{T})\geq 0$ for every $N$-tuple $\mathbf{T}$ of $n\times n$ matrices $\mathbf{T}\in\mathcal{C}^N\cap {\rm Nilp}_N(\Lambda_m),\ n\in\mathbb{N}$, where $m=\max_{w\in\Lambda}|w|$, is necessary for the solvability of Problem~\ref{prob:c-gen}?  Which admissible sets $\Lambda\subset\mathcal{F}_N$ are maximal in the sense that, for a certain choice of problem data $c_w,\ w\in\Lambda$, the condition above fails not only for $\Lambda_m\supset\Lambda$ but also for every admissible set $\tilde{\Lambda}\supset\Lambda$?

\section{The Carath\'{e}odory--Fej\'{e}r interpolation problem}\label{sec:prob-cf}
Recall that the \emph{non-commutative Schur--Agler class} $\mathcal{SA}_N^{\rm nc}(\mathcal{U,Y})$ consists of formal power series $F(z)=\sum_{w\in\mathcal{F}_N}F_wz^w\in\mathcal{L(U,Y)}\left\langle \left\langle z_1,\ldots,z_N\right\rangle\right\rangle$ such that  for every $\mathbf{C}\in\mathcal{D}^N$ (or equivalently, for every $\mathbf{C}\in\mathcal{D}^N_{\rm matr}$, see \cite{AK2})  the series $\sum_{w\in\mathcal{F}_N}F_w\otimes\mathbf{C}^w$ converges in the operator norm to the contractive operator $F(\mathbf{C})$.

Let us pose now the \emph{Carath\'{e}odory--Fej\'{e}r  problem in the class $\mathcal{SA}_N^{\rm nc}(\mathcal{U,Y})$}.
\begin{prob}\label{prob:cf}
Let $\Lambda\subset\mathcal{F}_N$ be an admissible set. Given a collection of operators $\{ s_w\}_{w\in\Lambda}\in\mathcal{L(U,Y)}$, find 
$F\in\mathcal{SA}_N^{\rm nc}(\mathcal{U,Y})$ such that
$$
F_w=s_w,\quad w\in\Lambda.
$$
\end{prob}
\begin{thm}\label{thm:cf}
Problem~\ref{prob:cf} has a solution if and only if the polynomial 
\begin{equation}\label{q-poly}
q(z):=\sum_{w\in\Lambda}s_wz^w
\end{equation}
satisfies $\| q(\mathbf{T})\|\leq 1$ for every $N$-tuple $\mathbf{T}$ of $\Lambda$-jointly nilpotent contractive operators. Moreover, for the solvability of Problem~\ref{prob:cf} it is enough to assume that $\| q(\mathbf{T})\|\leq 1$ holds for every $N$-tuple $\mathbf{T}$ of $\Lambda$-jointly nilpotent contractive $n\times n$ matrices, for all $n\in\mathbb{N}$.
\end{thm}
\begin{proof}
If Problem~\ref{prob:cf} has a solution $F\in\mathcal{SA}_N^{\rm nc}(\mathcal{Y})$ then  for any $\mathbf{C}\in\mathcal{D}^N\cap\mathcal{L(E)}^N$, with a Hilbert space $\mathcal{E}$, the series $F(\mathbf{C}):=\sum_{w\in\mathcal{F}_N}F_w\otimes\mathbf{C}^w$
converges in the operator norm, and $\| F(\mathbf{C})\|\leq 1$. If $\mathbf{T}\in\mathcal{C}^N\cap\mathcal{L(E)}^N$ is an $N$-tuple of $\Lambda$-jointly nilpotent operators then so is $r\mathbf{T}=(rT_1,\ldots,rT_N)\in\mathcal{D}^N\cap\mathcal{L(E)}^N$ for every $r:\ 0<r<1$. Therefore, $$\| q(r\mathbf{T})\|=\| F(r\mathbf{T})\|=\left\|\sum_{w\in\Lambda}s_w\otimes(r\mathbf{T})^w\right\|\leq 1.$$
By letting $r\uparrow 1$, we obtain $\| q(\mathbf{T})\|\leq 1$.

For the converse direction, let us consider first the case where $\mathcal{U}=\mathcal{Y}$ and $-I_\mathcal{Y}\leq s_\emptyset=s_\emptyset^*\leq 0$. Then the operator $I_\mathcal{Y}-s_\emptyset$ is boundedly invertible, and $(I_\mathcal{Y}+s_\emptyset)(I_\mathcal{Y}-s_\emptyset)^{-1}\geq 0$. Moreover, a formal power series $h(z):=(I_\mathcal{Y}+q(z))(I_\mathcal{Y}-q(z))^{-1}\in\mathcal{L(Y)}\left\langle \left\langle z_1,\ldots,z_N\right\rangle\right\rangle$ is well defined. Suppose that $\| q(\mathbf{T})\|\leq 1$ holds for every $N$-tuple $\mathbf{T}$ of $\Lambda$-jointly nilpotent contractive $n\times n$ matrices, for all $n\in\mathbb{N}$. Then
$\Re h(\mathbf{T})\geq 0$ for such a $\mathbf{T}$ (here $ h(\mathbf{T})=\sum_{w\in\Lambda}h_w\otimes\mathbf{T}^w$ is well defined). Define $c_\emptyset:=2h_\emptyset=2(I_\mathcal{Y}+s_\emptyset)(I_\mathcal{Y}-s_\emptyset)^{-1}\geq 0$, $c_w:=h_w$ for $w\in\Lambda\setminus\{\emptyset\}$. For the polynomial $p(z)$ defined by \eqref{poly} the condition that 
$\Re p(\mathbf{T})$ ($=\Re h(\mathbf{T})$) is a positive semidefinite operator for every $N$-tuple $\mathbf{T}$ of $\Lambda$-jointly nilpotent contractive $n\times n$ matrices, for all $n\in\mathbb{N}$, is fulfilled. By Theorem~\ref{thm:main}, there exists $f\in\mathcal{HA}_N^{\rm nc}(\mathcal{Y})$ such that $f_\emptyset=\frac{c_\emptyset}{2}=h_\emptyset$, $f_w=c_w=h_w$ for $w\in\Lambda\setminus\{\emptyset\}$. Then the formal power series $F(z):=(f(z)-I_\mathcal{Y})(f(z)+I_\mathcal{Y})^{-1}\in\mathcal{L(Y)}\left\langle \left\langle z_1,\ldots,z_N\right\rangle\right\rangle$ is well defined and belongs to the class $\mathcal{SA}_N^{\rm nc}(\mathcal{Y})$ (see Section~\ref{sec:ha}). Since $f(\mathbf{T})=h(\mathbf{T})=p(\mathbf{T})$  for any $N$-tuple $\mathbf{T}$ of $\Lambda$-jointly nilpotent contractive $n\times n$ matrices,  $n\in\mathbb{N}$, we get $F(\mathbf{T})=q(\mathbf{T})$ for  such a $\mathbf{T}$. Thus, the polynomial
$\sum_{w\in\Lambda}(F_w-s_w)z^w\in\mathcal{L(Y)}\left\langle z_1,\ldots,z_N\right\rangle$ vanishes on $N$-tuples of $\Lambda$-jointly nilpotent contractive $n\times n$ matrices, for every $n\in\mathbb{N}$. By Proposition~\ref{prop:vanish},
$F_w=s_w$ for all $w\in\Lambda$, i.e., $F$ solves Problem~\ref{prob:cf} for the data $s_w,\ w\in\Lambda$. 

Consider now the case where $\mathcal{U}=\mathcal{Y}$, however $s_\emptyset$ is not necessarily selfadjoint and negative semidefinite. The operator $s_\emptyset$ has a polar deconposition $s_\emptyset=UR$, where $U\in\mathcal{L(Y)}$ is unitary and $R\in\mathcal{L(Y)}$ is positive semidefinite. Suppose that $\| q(\mathbf{T})\|\leq 1$ holds for every $N$-tuple $\mathbf{T}$ of $\Lambda$-jointly nilpotent contractive $n\times n$ matrices, for all $n\in\mathbb{N}$. Then $s_\emptyset$ is a contraction, and $-I_\mathcal{Y}\leq-R\leq 0$. Define the operators $\widetilde{s_w}:=-U^*s_w,\ w\in\Lambda,$ and the polynomial $\widetilde{q}(z):=-U^*q(z)$. Clearly,  $\| \widetilde{q}(\mathbf{T})\|\leq 1$ holds for every $N$-tuple $\mathbf{T}$ of $\Lambda$-jointly nilpotent contractive $n\times n$ matrices, for all $n\in\mathbb{N}$, and $-I_\mathcal{Y}\leq\widetilde{s_\emptyset}=-R\leq 0$. By the result of the previous paragraph, there exists a solution $\widetilde{F}(z)\in\mathcal{SA}_N^{\rm nc}(\mathcal{Y})$ to Problem~\ref{prob:cf} for the data $\widetilde{s_w},\ w\in\Lambda$. Then $F(z):=-U\widetilde{F}(z)\in\mathcal{SA}_N^{\rm nc}(\mathcal{Y})$ is a solution to Problem~\ref{prob:cf} for the data $s_w,\ w\in\Lambda$.

Consider now the case where  $\mathcal{U}$ does not necessarily coincide with $\mathcal{Y}$.  Suppose that $\| q(\mathbf{T})\|\leq 1$ holds for every $N$-tuple $\mathbf{T}$ of $\Lambda$-jointly nilpotent contractive $n\times n$ matrices, for all $n\in\mathbb{N}$. Define \begin{eqnarray*}\widetilde{s_w} &:=& \left[\begin{array}{cc}
0 & 0\\
s_w & 0\end{array}\right]\in\mathcal{L(U\oplus Y)},\quad w\in\Lambda,\\
 \widetilde{q}(z) &:=& \left[\begin{array}{cc}
0 & 0\\
q(z) & 0\end{array}\right]\in\mathcal{L(U\oplus Y)}\left\langle z_1,\ldots,z_N\right\rangle.
\end{eqnarray*}
Clearly, $\| \widetilde{q}(\mathbf{T})\|\leq 1$ holds for every $N$-tuple $\mathbf{T}$ of $\Lambda$-jointly nilpotent contractive $n\times n$ matrices, for all $n\in\mathbb{N}$. By the result of the previous paragraph, there exists a solution $\widetilde{F}(z)\in\mathcal{SA}_N^{\rm nc}(\mathcal{U\oplus Y})$ to Problem~\ref{prob:cf} for the data $\widetilde{s_w},\ w\in\Lambda$. Then $F(z):=P_\mathcal{Y}\widetilde{F}(z)\big|_\mathcal{U}\in\mathcal{SA}_N^{\rm nc}(\mathcal{U,Y})$ is a solution to Problem~\ref{prob:cf} for the data $s_w,\ w\in\Lambda$.
\end{proof}
\begin{rem}
The referee suggested that the fact that the quotient of an operator algebra by an ideal
 is itself an operator algebra (see \cite{BRS}) may provide an alternate approach to proving Theorem~\ref{thm:cf}. We leave it now for a possible further exploration.
\end{rem}
Let us remark also that the examples analogous to Examples~\ref{ex:1} and \ref{ex:2} can be easily constructed for the setting of the present section, and one can ask the following questions. For which admissible sets $\Lambda\subset\mathcal{F}_N$ and data $s_w,\ w\in\Lambda$, the condition that $\| q(\mathbf{T})\|\leq 1$ for every $N$-tuple $\mathbf{T}$ of $n\times n$ matrices $\mathbf{T}\in\mathcal{C}^N\cap {\rm Nilp}_N(\Lambda_m),\ n\in\mathbb{N}$, where $m=\max_{w\in\Lambda}|w|$, is necessary for the solvability of Problem~\ref{prob:cf}?  Which admissible sets $\Lambda\subset\mathcal{F}_N$ are maximal in the sense that, for a certain choice of problem data $s_w,\ w\in\Lambda$, the condition above fails not only for $\Lambda_m\supset\Lambda$ but also for every admissible set $\tilde{\Lambda}\supset\Lambda$?
\section*{Aknowledgements}

I am thankful to Prof. Hugo Woerdeman for discussions which stimulated my interest to non-commutative interpolation problems. I express my gratitude to Prof. Victor Katsnelson for a copy of his unpublished monograph \cite{Kats} which is a good source of mathematical and historical information on classical interpolation problems and other related questions in analysis.

%\bibliographystyle{plain}
%\bibliography{caratheo}

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