---
abstract: 'Let $t\mapsto A(t)$ for $t\in T$ be a $C^M$-mapping with values unbounded operators with compact resolvents and common domain of definition which are self-adjoint or normal. Here $C^M$ stands for $C^\omega$ (real analytic), a quasianalytic or non-quasianalytic Denjoy-Carleman class, $C^\infty$, or a Hölder continuity class $C^{0,\alpha}$. The parameter domain $T$ is either $\mathbb R$ or $\mathbb R^n$ or an infinite dimensional convenient vector space. We prove and review results on $C^M$-dependence on $t$ of the eigenvalues and eigenvectors of $A(t)$.'
address:
- 'Andreas Kriegl: Fakultät für Mathematik, Universität Wien, Nordbergstrasse 15, A-1090 Wien, Austria'
- 'Peter W. Michor: Fakultät für Mathematik, Universität Wien, Nordbergstrasse 15, A-1090 Wien, Austria'
- 'Armin Rainer: Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, Ontario, Canada M5S 2E4'
author:
- 'Andreas Kriegl, Peter W. Michor, and Armin Rainer'
title: 'Denjoy-Carleman differentiable perturbation of polynomials and unbounded operators '
---

[Theorem]{} Let $t\mapsto A(t)$ for $t\in T$ be a parameterized family of unbounded operators in a Hilbert space $H$ with common domain of definition and with compact resolvent.

If $t\in T=\mathbb R$ and all $A(t)$ are self-adjoint then the following holds:

1.  If $A(t)$ is real analytic in $t\in \mathbb R$, then the eigenvalues and the eigenvectors of $A(t)$ may be parameterized real analytically in $t$.

2.  If $A(t)$ is quasianalytic of class $C^Q$ in $t\in \mathbb R$, then the eigenvalues and the eigenvectors of $A(t)$ may be parameterized $C^Q$ in $t$.

3.  If $A(t)$ is non-quasianalytic of class $C^L$ in $t\in \mathbb R$ and if no two unequal continuously parameterized eigenvalues meet of infinite order at any $t\in \mathbb R$, then the eigenvalues and the eigenvectors of $A(t)$ can be parameterized $C^L$ in $t$.

4.  If $A(t)$ is $C^\infty$ in $t\in \mathbb R$ and if no two unequal continuously parameterized eigenvalues meet of infinite order at any $t\in \mathbb R$, then the eigenvalues and the eigenvectors of $A(t)$ can be parameterized $C^\infty$ in $t$.

5.  If $A(t)$ is $C^\infty$ in $t\in \mathbb R$, then the eigenvalues of $A(t)$ may be parameterized twice differentiably in $t$.

6.  If $A(t)$ is $C^{1,\alpha}$ in $t\in \mathbb R$ for some $\alpha>0$, then the eigenvalues of $A(t)$ may be parameterized in a $C^1$ way in $t$.

If $t\in T=\mathbb R$ and all $A(t)$ are normal then the following holds:

1.  If $A(t)$ is real analytic in $t\in \mathbb R$, then for each $t_0\in \mathbb R$ and for each eigenvalue $\lambda$ of $A(t_0)$ there exists $N\in \mathbb N$ such that the eigenvalues near $\lambda$ of $A(t_0\pm s^N)$ and their eigenvectors can be parameterized real analytically in $s$ near $s=0$.

2.  If $A(t)$ is $C^Q$ in $t\in \mathbb R$, then for each $t_0\in \mathbb R$ and for each eigenvalue $\lambda$ of $A(t_0)$ there exists $N\in \mathbb N$ such that the eigenvalues near $\lambda$ of $A(t_0\pm s^N)$ and their eigenvectors can be parameterized $C^Q$ in $s$ near $s=0$.

3.  If $A(t)$ is $C^L$ in $t\in \mathbb R$, then for each $t_0\in \mathbb R$ and for each eigenvalue $\lambda$ of $A(t_0)$ at which no two of the unequal continuously arranged eigenvalues (see [@Kato76 II.5.2]) meet of infinite order, there exists $N\in \mathbb N$ such that the eigenvalues near $\lambda$ of $A(t_0\pm s^N)$ and their eigenvectors can be parameterized $C^L$ in $s$ near $s=0$.

4.  If $A(t)$ is $C^\infty$ in $t\in \mathbb R$, then for each $t_0\in \mathbb R$ and for each eigenvalue $\lambda$ of $A(t_0)$ at which no two of the unequal continuously arranged eigenvalues (see [@Kato76 II.5.2]) meet of infinite order, there exists $N\in \mathbb N$ such that the eigenvalues near $\lambda$ of $A(t_0\pm s^N)$ and their eigenvectors can be parameterized $C^\infty$ in $s$ near $s=0$.

5.  If $A(t)$ is $C^\infty$ in $t\in \mathbb R$ and no two of the unequal continuously parameterized eigenvalues meet of infinite order at any $t\in \mathbb R$, then the eigenvalues and the eigenvectors of $A(t)$ can be parameterized by absolutely continuous functions, locally in $t$.

If $t\in T=\mathbb R^n$ and all $A(t)$ are normal then the following holds:

1.  If $A(t)$ is $C^\omega$ or $C^Q$ in $t\in \mathbb R^n$, then for each $t_0\in \mathbb R^n$ and for each eigenvalue $\lambda$ of $A(t_0)$, there exist a finite covering $\{\pi_k : U_k \to W\}$ of a neighborhood $W$ of $t_0$, where each $\pi_k$ is a composite of finitely many mappings each of which is either a local blow-up along a $C^\omega$ or $C^Q$ submanifold or a local power substitution, such that the eigenvalues and the eigenvectors of $A(\pi_k(s))$ can be chosen $C^\omega$ or $C^Q$ in $s$. If $A$ is self-adjoint, then we do not need power substitutions.

2.  If $A(t)$ is $C^\omega$ or $C^Q$ in $t\in \mathbb R^n$, then the eigenvalues and their eigenvectors of $A(t)$ can be parameterized by functions which are special functions of bounded variation (SBV), see [@AmbrosioDeGiorgi88] or [@AFP00], locally in $t$.

If $t\in T\subseteq E$, a $c^\infty$-open subset in an infinite dimensional convenient vector space then the following holds:

1.  For $0<\alpha\le 1$, if $A(t)$ is $C^{0,\alpha}$ (Hölder continuous of exponent $\alpha$) in $t\in T$ and all $A(t)$ are self-adjoint, then the eigenvalues of $A(t)$ may be parameterized in a $C^{0,\alpha}$ way in $t$.

2.  For $0<\alpha\le 1$, if $A(t)$ is $C^{0,\alpha}$ (Hölder continuous of exponent $\alpha$) in $t\in T$ and all $A(t)$ are normal, then we have: For each $t_0\in T$ and each eigenvalue $z_0$ of $A(t_0)$ consider a simple closed $C^1$-curve $\gamma$ in the resolvent set of $A(t_0)$ enclosing only $z_0$ among all eigenvalues of $A(t_0)$. Then for $t$ near $t_0$ in the $c^\infty$-topology on $T$, no eigenvalue of $A(t)$ lies on $\gamma$. Let $\lambda(t)=(\lambda_1(t),\dots,\lambda_N(t))$ be the $N$-tuple of all eigenvalues (repeated according to their multiplicity) of $A(t)$ inside of $\gamma$. Then $t\mapsto \lambda(t)$ is $C^{0,\alpha}$ for $t$ near $t_0$ with respect to the non-separating metric $$d(\lambda,\mu) = \min_{\sigma\in\mathcal S_N} \max_{1\le i\le N} |\lambda_i - \mu_{\sigma(i)}|$$ on the space of $N$-tuples.

Part [(A)]{} is due to Rellich [@Rellich42V] in 1942, see also [@Baumgaertel72] and [@Kato76 VII, 3.9]. Part [(D)]{} has been proved in [@AKLM98 7.8], see also [@KM97 50.16], in 1997, which contains also a different proof of [(A)]{}. [(E)]{} and [(F)]{} have been proved in [@KM03] in 2003. (G) was proved in [@RainerAC 7.1]; it can be proved as [(H)]{} with some obvious changes, but it is not a special case since $C^\omega$ does not correspond to a sequence which is an $\mathcal L$-intersection (see [@KMRq]). [(J)]{} and [(K)]{} were proved in [@RainerAC 7.1]. [(N)]{} was proved in [@KMR].

The purpose of this paper is to prove the remaining parts , , , , , , and .

Definitions and remarks  {#definitions-and-remarks .unnumbered}
------------------------

Let $M=(M_k)_{k \in \mathbb{N}=\mathbb{N}_{\ge0}}$ be an increasing sequence ($M_{k+1}\ge M_k$) of positive real numbers with $M_0=1$. Let $U \subseteq \mathbb{R}^n$ be open. We denote by $C^M(U)$ the set of all $f \in C^\infty(U)$ such that, for each compact $K \subseteq U$, there exist positive constants $C$ and $\rho$ such that $$|\partial^\alpha f(x)| \le C \, \rho^{|\alpha|} \, |\alpha|! \, M_{|\alpha|}
\quad\text{ for all }\alpha \in \mathbb{N}^n\text{ and }x \in K.$$ The set $C^M(U)$ is a *Denjoy–Carleman class* of functions on $U$. If $M_k=1$, for all $k$, then $C^M(U)$ coincides with the ring $C^\omega(U)$ of real analytic functions on $U$. In general, $C^\omega(U) \subseteq C^M(U) \subseteq C^\infty(U)$.

Here $Q=(Q_k)_{k\in\mathbb N}$ is a sequence as above which is quasianalytic, log-convex, and which is also an $\mathcal L$-intersection, see [@KMRq] or [@KMRc] and references therein. Moreover, $L=(L_k)_{k\in\mathbb N}$ is a sequence as above which is non-quasianalytic and log-convex.

That $A(t)$ is a real analytic, $C^M$ (where $M$ is either $Q$ or $L$), $C^\infty$, or $C^{k,\alpha}$ family of unbounded operators means the following: There is a dense subspace $V$ of the Hilbert space $H$ such that $V$ is the domain of definition of each $A(t)$, and such that $A(t)^*=A(t)$ in the self-adjoint case, or $A(t)$ has closed graph and $A(t)A(t)^*=A(t)^*A(t)$ wherever defined in the normal case. Moreover, we require that $t\mapsto \langle A(t)u,v\rangle$ is of the respective differentiability class for each $u\in V$ and $v\in H$. &gt;From now on we treat only $C^M=C^\omega$, $C^M$ for $M=Q$, $M=L$, and $C^M=C^{0,\alpha}$.

This implies that $t\mapsto A(t)u$ is of the same class $C^M(E,H)$ (where $E$ is either $\mathbb R$ or $\mathbb R^n$) or is in $C^{0,\alpha}(E,H)$ (if $E$ is a convenient vector space) for each $u\in V$ by [@KM97 2.14.4, 10.3] for $C^\omega$, by [@KMRc 3.1, 3.3, 3.5] for $M=L$, by [@KMRq 1.10, 2.1, 2.3] for $M=Q$, and by [@KM97 2.3], [@FK88 2.6.2] or [@Faure91 4.14.4] for $C^{0,\alpha}$ because $C^{0,\alpha}$ can be described by boundedness conditions only and for these the uniform boundedness principle is valid.

A sequence of functions $\lambda_i$ is said to [*parameterize the eigenvalues, if for each $z\in \mathbb C$ the cardinality $|\{i: \lambda_i(t)=z\}|$ equals the multiplicity of $z$ as eigenvalue of $A(t)$.*]{}

Let $X$ be a $C^\omega$ or $C^Q$ manifold. A *local blow-up $\Phi$* over an open subset $U$ of $X$ means the composition $\Phi = \iota \circ \varphi$ of a blow-up $\varphi : U' \to U$ with center a $C^\omega$ or $C^Q$ submanifold and of the inclusion $\iota : U \to X$. A *local power substitution* is a mapping $\Psi: V \to X$ of the form $\Psi = \iota \circ \psi$, where $\iota : W \to X$ is the inclusion of a coordinate chart $W$ of $X$ and $\psi : V \to W$ is given by $$(y_1,\ldots,y_q) = ((-1)^{\epsilon_1} x_1^{\gamma_1},\ldots,(-1)^{\epsilon_q} x_q^{\gamma_q}),$$ for some $\gamma=(\gamma_1,\ldots,\gamma_q) \in (\mathbb{N}_{>0})^q$ and all $\epsilon = (\epsilon_1,\ldots,\epsilon_q) \in \{0,1\}^q$, where $y_1,\ldots,y_q$ denote the coordinates of $W$ (and $q = \dim X$).

This paper became possible only after some of the results of [@KMRc] and [@KMRq] were proved, in particular the uniform boundedness principles. The wish to prove the results of this paper was the main motivation for us to work on [@KMRc] and [@KMRq].

Applications  {#applications .unnumbered}
-------------

Let $X$ be a compact $C^Q$ manifold and let $t\mapsto g_t$ be a $C^Q$-curve of $C^Q$ Riemannian metrics on $X$. Then we get the corresponding $C^Q$ curve $t\mapsto \Delta(g_t)$ of Laplace-Beltrami operators on $L^2(X)$. By theorem (B) the eigenvalues and eigenvectors can be arranged $C^Q$. Question: Are the eigenfunctions then also $C^Q$?

Let $\Omega$ be a bounded region in $\mathbb R^n$ with $C^Q$ boundary, and let $H(t)=-\Delta + V(t)$ be a $C^Q$-curve of Schrödinger operators with varying $C^Q$ potential and Dirichlet boundary conditions. Then the eigenvalues and eigenvectors can be arranged $C^Q$. Question: Are the eigenvectors viewed as eigenfunctions then also in $C^Q(\Omega\times \mathbb R)$?

Example  {#example .unnumbered}
--------

This is an elaboration of [@AKLM98 7.4] and [@KM03 Example]. Let $S(2)$ be the vector space of all symmetric real $(2\times 2)$-matrices. We use the $C^L$-curve lemma [@KMRc 3.6] or [@KMRq 2.5]: [*There exists a converging sequence of reals $t_n$ with the following property: Let $A_n, B_n\in S(2)$ be any sequences which converge fast to 0, i.e., for each $k\in \mathbb N$ the sequences $n^kA_n$ and $n^kB_n$ are bounded in $S(2)$. Then there exists a curve $A\in C^L(\mathbb R,S(2))$ such that $A(t_n+s)=A_n+sB_n$ for $|s|\le \frac1{n^2}$, for all $n$.*]{}

We use it for $$\begin{aligned}
A_n := \frac 1{2^{n^2}} 
\begin{pmatrix}
1 & 0 \\
0 & -1\\
\end{pmatrix},\quad 
B_n := \frac 1{2^{n^2}\,s_n} 
\begin{pmatrix}
0 & 1 \\
1 & 0\\
\end{pmatrix},\quad 
\text{ where } s_n := 2^{n-n^2}\le \frac 1{n^2}.\end{aligned}$$ The eigenvalues of $A_n+tB_n$ and their derivatives are $$\lambda_n(t) = \pm\frac 1{2^{n^2}} \sqrt{1+(\tfrac t{s_n})^2},\quad
\lambda_n'(t) = \pm\frac {2^{n^2-2n}t}{\sqrt{1+(\frac t{s_n})^2}}.$$ Then $$\begin{aligned}
\frac{\lambda'(t_n+s_n)-\lambda'(t_n)}{s_n^\alpha} &=
\frac{\lambda_n'(s_n)-\lambda_n'(0)}{s_n^\alpha}
=\pm\frac {2^{n^2-2n}s_n}{s_n^\alpha\sqrt{2}}\\
&=\pm\frac{2^{n(\alpha(n-1)-1)}}{\sqrt{2}} \to \infty \text{ for }\alpha>0.\end{aligned}$$ So condition (in , , , , and ) that no two unequal continuously parameterized eigenvalues meet of infinite order cannot be dropped. By [@AKLM98 2.1], we may always find a twice differentiable square root of a non-negative smooth function, so that the eigenvalues $\lambda$ are functions which are twice differentiable but not $C^{1,\alpha}$ for any $\alpha>0$.

Note that the normed eigenvectors cannot be chosen continuously in this example (see also example [@Rellich37I §2]). Namely, we have $$A(t_n)=A_n=\frac1{2^{n^2}}\begin{pmatrix} 1 & 0 \\ 0 &-1 \end{pmatrix},\qquad
A(t_n+s_n)=A_n +s_n\,B_n=\frac1{2^{n^2}}\begin{pmatrix} 1 & 1 \\ 1 &-1 \end{pmatrix}.$$

[Resolvent Lemma]{} Let $C^M$ be any of $C^\omega$, $C^Q$, $C^L$, $C^\infty$, or $C^{0,\alpha}$, and let $A(t)$ be normal. If $A$ is $C^M$ then the resolvent $(t,z)\mapsto (A(t)-z)^{-1}\in L(H,H)$ is $C^M$ on its natural domain, the global resolvent set $$\{(t,z)\in T\times\mathbb C: (A(t)-z):V\to H \text{ is invertible}\}$$ which is open (and even connected).

[**Proof**]{} By definition the function $t\mapsto \langle A(t)v,u \rangle$ is of class $C^M$ for each $v\in V$ and $u\in H$. We may conclude that the mapping $t\mapsto A(t)v$ is of class $C^M$ into $H$ as follows: For $C^M=C^\infty$ we use [@KM97 2.14.4]. For $C^M=C^\omega$ we use in addition [@KM97 10.3]. For $C^M=C^Q$ or $C^M=C^L$ we use [@KMRq 2.1] and/or [@KMRc 3.3] where we replace $\mathbb R$ by $\mathbb R^n$. For $C^M=C^{0,\alpha}$ we use [@KM97 2.3], [@FK88 2.6.2], or [@Faure91 4.1.14] because $C^{0,\alpha}$ can be described by boundedness conditions only and for these the uniform boundedness principle is valid.

For each $t$ consider the norm $\|u\|_t^2:=\|u\|^2+\|A(t)u\|^2$ on $V$. Since $A(t)$ is closed, $(V,\|\quad\|_t)$ is again a Hilbert space with inner product $\langle u,v\rangle_t:=\langle u,v\rangle+\langle A(t)u,A(t)v\rangle$.

[*[(1)]{} Claim (see [@AKLM98 in the proof of 7.8], [@KM97 in the proof of 50.16], or [@KM03 Claim 1]). All these norms $\|\quad\|_t$ on $V$ are equivalent, locally uniformly in $t$. We then equip $V$ with one of the equivalent Hilbert norms, say $\|\quad\|_0$.*]{}

We reduce this to $C^{0,\alpha}$. Namely, note first that $A(t):(V,\|\quad\|_s)\to H$ is bounded since the graph of $A(t)$ is closed in $H\times H$, contained in $V\times H$ and thus also closed in $(V,\|\quad\|_s)\times H$. For fixed $u,v\in V$, the function $t\mapsto \langle u,v\rangle_t=\langle u,v \rangle+\langle A(t)u,A(t)v 
\rangle$ is $C^{0,\alpha}$ since $t\mapsto A(t)u$ is it. By the multilinear uniform boundedness principle ([@KM97 5.18] or [@FK88 3.7.4]) the mapping $t\mapsto \langle \quad,\quad\rangle_t$ is $C^{0,\alpha}$ into the space of bounded sesquilinear forms on $(V,\|\quad\|_s)$ for each fixed $s$. Thus the inverse image of $\langle \quad,\quad \rangle_s + \frac12(\text{unit ball})$ in $L(\overline{(V,\|\quad\|_s)} \oplus (V,\|\quad\|_s);\mathbb C)$ is a $c^\infty$-open neighborhood $U$ of $s$ in $T$. Thus $\sqrt{1/2}\|u\|_s\le \|u\|_t\le \sqrt{3/2}\|u\|_s$ for all $t\in U$, i.e.  all Hilbert norms $\|\quad\|_t$ are locally uniformly equivalent, and claim [(1)]{} follows.

By the linear uniform boundedness theorem we see that $t\mapsto A(t)$ is in $C^M(T, L(V,H))$ as follows (here it suffices to use a set of linear functionals which together recognize bounded sets instead of the whole dual): For $C^M=C^\infty$ we use [@KM97 1.7 and 2.14.3]. For $C^M=C^\omega$ we use in addition [@KM97 9.4]. For $C^M=C^Q$ or $C^M=C^L$ we use [@KMRq 2.2 and 2.3] and/or [@KMRc 3.5] where we replace $\mathbb R$ by $\mathbb R^n$. For $C^M=C^{0,\alpha}$ see above.

If for some $(t,z)\in T\times\mathbb C$ the bounded operator $A(t)-z:V\to H$ is invertible, then this is true locally with respect to the $c^\infty$-topology on the product which is the product topology by [@KM97 4.16], and $(t,z)\mapsto (A(t)-z)^{-1}:H\to V$ is $C^M$, by the chain rule, since inversion is real analytic on the Banach space $L(V,H)$.

Note that $(A(t)-z)^{-1}:H\to H$ is a compact operator for some (equivalently any) $(t,z)$ if and only if the inclusion $i:V\to H$ is compact, since $i=(A(t)-z)^{-1}\circ(A(t)-z): V\to H\to H$.

[Polynomial proposition]{} Let $P$ be a curve of polynomials $$P(t)(x)=x^n-a_1(t)x^{n-1}+\dots+(-1)^na_n(t),\quad t\in \mathbb R.$$

1.  If $P$ is hyperbolic (all roots real) and if the coefficient functions $a_i$ are all $C^Q$ then there exist $C^Q$ functions $\lambda_i$ which parameterize all roots.

2.  If $P$ is hyperbolic (all roots real), if the coefficient functions $a_i$ are $C^L$ and no two of the different roots meet of infinite order, then there exist $C^L$ functions $\lambda_i$ which parameterize all roots.

3.  If the coefficient functions $a_i$ are $C^Q$, then for each $t_0$ there exists $N\in \mathbb N$ such that the roots of $s\mapsto P(t_0\pm s^{N})$ can be parameterized $C^Q$ in $s$ for $s$ near 0.

4.  If the coefficient functions $a_i$ are $C^L$ and no two of the different roots meet of infinite order, then for each $t_0$ there exists $N\in \mathbb N$ such that the roots of $s\mapsto P(t_0\pm s^{N})$ can be parameterized $C^L$ in $s$ for $s$ near 0.

All $C^Q$ or $C^L$ solutions differ by permutations.

The proof of parts and is exactly as in [@AKLM98] where the corresponding results were proven for $C^\infty$ instead of $C^L$, and for $C^\omega$ instead of $C^Q$. For this we need only the following properties of $C^Q$ and $C^L$:

-   They allow for the implicit function theorem (for [@AKLM98 3.3]).

-   They contain $C^\omega$ and are closed under composition (for [@AKLM98 3.4]).

-   They are derivation closed (for [@AKLM98 3.7]).

Part is also in [@CC04 7.6] which follows [@AKLM98]. It also follows from the multidimensional version [@RainerQA 6.10] since blow-ups in dimension 1 are trivial. The proofs of parts and are exactly as in [@RainerAC 3.2] where the corresponding result was proven for $C^\omega$ instead of $C^Q$, and for $C^\infty$ instead of $C^L$, if none of the different roots meet of infinite order. For these we need the properties of $C^Q$ and $C^L$ listed above.

[Matrix proposition]{} Let $A(t)$ for $t\in T$ be a family of $(N\times N)$-matrices.

1.  If $T=\mathbb R\ni t\mapsto A(t)$ is a $C^Q$-curve of Hermitian matrices, then the eigenvalues and the eigenvectors can be chosen $C^Q$.

2.  If $T=\mathbb R\ni t\mapsto A(t)$ is a $C^L$-curve of Hermitian matrices such that no two eigenvalues meet of infinite order, then the eigenvalues and the eigenvectors can be chosen $C^L$.

3.  If $T=\mathbb R\ni t\mapsto A(t)$ is a $C^L$-curve of normal matrices such that no two eigenvalues meet of infinite order, then for each $t_0$ there exists $N_1\in \mathbb N$ such that the eigenvalues and eigenvectors of $s\mapsto A(t_0\pm s^{N_1})$ can be parameterized $C^L$ in $s$ for $s$ near 0.

4.  Let $T\subseteq\mathbb R^n$ be open and let $T\ni t\mapsto A(t)$ be a $C^\omega$ or $C^Q$-mapping of normal matrices. Let $K \subseteq T$ be compact. Then there exist a neighborhood $W$ of $K$, and a finite covering $\{\pi_k : U_k \to W\}$ of $W$, where each $\pi_k$ is a composite of finitely many mappings each of which is either a local blow-up along a $C^\omega$ or $C^Q$ submanifold or a local power substitution, such that the eigenvalues and the eigenvectors of $A(\pi_k(s))$ can be chosen $C^\omega$ or $C^Q$ in $s$. Consequently, the eigenvalues and eigenvectors of $A(t)$ are locally special functions of bounded variation (SBV). If $A$ is a family of Hermitian matrices, then we do not need power substitutions.

The proof of the matrix proposition in case and is exactly as in [@AKLM98 7.6], using the polynomial proposition and properties of $C^Q$ and $C^L$. Item is exactly as in [@RainerAC 6.2], using the polynomial proposition and properties of $C^L$. Item is proved in [@RainerQA 9.1 and 9.6], see also [@KurdykaPaunescu08].

[**Proof of the theorem**]{} We have to prove parts , , , , , , and . So let $C^M$ be any of $C^\omega$, $C^Q$, $C^L$, or $C^{0,\alpha}$, and let $A(t)$ be normal. Let $z$ be an eigenvalue of $A(t_0)$ of multiplicity $N$. We choose a simple closed $C^1$ curve $\gamma$ in the resolvent set of $A(t_0)$ for fixed $t_0$ enclosing only $z$ among all eigenvalues of $A(t_0)$. Since the global resolvent set is open, see the resolvent lemma, no eigenvalue of $A(t)$ lies on $\gamma$, for $t$ near $t_0$. By the resolvent lemma, $A: T\to L((V,\|\quad\|_0),H)$ is $C^M$, thus also $$t\mapsto -\frac1{2\pi i}\int_\gamma (A(t)-z)^{-1}\;dz =: P(t,\gamma) = P(t)$$ is a $C^M$ mapping. Each $P(t)$ is a projection, namely onto the direct sum of all eigenspaces corresponding to eigenvalues of $A(t)$ in the interior of $\gamma$, with finite rank. Thus the rank must be constant: It is easy to see that the (finite) rank cannot fall locally, and it cannot increase, since the distance in $L(H,H)$ of $P(t)$ to the subset of operators of rank $\le N=\operatorname{rank}(P(t_0))$ is continuous in $t$ and is either 0 or 1.

So for $t$ in a neighborhood $U$ of $t_0$ there are equally many eigenvalues in the interior of $\gamma$, and we may call them $\lambda_i(t)$ for $1\le i\le N$ (repeated with multiplicity).

Now we consider the family of $N$-dimensional complex vector spaces $t\mapsto P(t)(H)\subseteq H$, for $t\in U$. They form a $C^M$ Hermitian vector subbundle over $U$ of $U\times H\to U$: For given $t$, choose $v_1,\dots v_N\in H$ such that the $P(t)v_i$ are linearly independent and thus span $P(t)H$. This remains true locally in $t$. Now we use the Gram Schmidt orthonormalization procedure (which is $C^\omega$) for the $P(t)v_i$ to obtain a local orthonormal $C^M$ frame of the bundle.

Now $A(t)$ maps $P(t)H$ to itself; in a $C^M$ local frame it is given by a normal $(N\times N)$-matrix parameterized $C^M$ by $t\in U$.

Now all local assertions of the theorem follow:

-   Use the matrix proposition, part .

-   Use the matrix proposition, part .

-   Use the matrix proposition, part , and note that in dimension 1 blowups are trivial.

-   Use the matrix proposition, part .

-   Use the matrix proposition, part , for $\mathbb R^n$.

-   We use the following

Finally, it remains to extend the local choices to global ones for the cases and only. There $t\mapsto A(t)$ is $C^Q$ or $C^L$, respectively, which imply both $C^\infty$, and no two different eigenvalues meet of infinite order. So we may apply [@AKLM98 7.8] (in fact we need only the end of the proof) to conclude that the eigenvalues can be chosen $C^\infty$ on $T=\mathbb R$, uniquely up to a global permutation. By the local result above they are then $C^Q$ or $C^L$. The same proof then gives us, for each eigenvalue $\lambda_i:T \to \mathbb R$ with generic multiplicity $N$, a unique $N$-dimensional smooth vector subbundle of $\mathbb R\times H$ whose fiber over $t$ consists of eigenvectors for the eigenvalue $\lambda_i(t)$. In fact this vector bundle is $C^Q$ or $C^L$ by the local result above, namely the matrix proposition, part or , respectively.

\[2\][ [\#2](http://www.ams.org/mathscinet-getitem?mr=#1) ]{} \[2\][\#2]{}

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