Garling sequence spaces
Ben Wallis
By generalizing a construction of Garling, for each $1\leqslant p<\infty$ and each normalized, nonincreasing sequence of positive numbers $w\in c_0\setminus\ell_1$ we exhibit an $\ell_p$-saturated, complementably homogeneous Banach space $g(w,p)$ related to the Lorentz sequence space $d(w,p)$. Using methods originally developed for studying $d(w,p)$, we show that $g(w,p)$ admits a unique (up to equivalence) subsymmetric basis, although when $w=(n^{-\theta)_{n=1^\infty$ for some $0<\theta<1$, it does not admit a symmetric basis. We then discuss some additional properties of $g(w,p)$ related to uniform convexity and superreflexivity.
1612.01145.tex
\documentclass[draft,12pt]{amsart}
\usepackage[all]{xy}
\usepackage{amscd}

%\documentclass[12pt]{book}  
%\usepackage{proc}       

\usepackage{latexsym}
%\usepackage{Delete}

\usepackage{amsfonts}
\usepackage{amssymb}




%Here's my standard list of good stuff...
\DeclareMathOperator{\arsinh}{arsinh}
\DeclareMathOperator{\arcosh}{arcosh}
\DeclareMathOperator{\spa}{span}
\DeclareMathOperator{\tr}{Tr}
\DeclareMathOperator{\re}{Re}
\DeclareMathOperator{\im}{Im}
\DeclareMathOperator{\id}{Id}
\DeclareMathOperator{\op}{Op}
\DeclareMathOperator{\su}{su}
\DeclareMathOperator{\rank}{Rank}
\DeclareMathOperator{\diam}{Diam}
\DeclareMathOperator{\supp}{Supp}
\DeclareMathOperator{\simp}{Simp}
\DeclareMathOperator{\meas}{Meas}
\DeclareMathOperator{\sgn}{Sgn}
\DeclareMathOperator{\codim}{codim}
\DeclareMathOperator{\esssup}{ess\,sup}
\DeclareMathOperator{\bv}{\emph{\textbf{v}}}
\DeclareMathOperator{\bw}{\emph{\textbf{w}}}
\DeclareMathOperator{\bve}{\textbf{v}}
\DeclareMathOperator{\bwe}{\textbf{w}}
\newcommand{\norm}[1]{\lVert#1\rVert}
\newcommand{\bignorm}[1]{\left\lVert#1\right\rVert}
\newcommand{\tuple}[1]{\langle#1\rangle}
\newcommand{\bigtuple}[1]{\left\langle#1\right\rangle}
%End standard list of good stuff.



\renewcommand{\leq}{\leqslant}
\renewcommand{\geq}{\geqslant}



%\swapnumbers

\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{conjecture}[theorem]{Conjecture}


\newtheorem{problem}[theorem]{Problem}

\theoremstyle{plain}

%\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{question}[theorem]{Question}
\newtheorem{example}[theorem]{Example}
\newtheorem{numbered}[theorem]{}


\newcommand{\marg}[1]{\marginpar{\tiny #1}}




%
%
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%
%               ***** Page Size *****
\topmargin=-0cm
%\textheight=25cm
%\textheight=25cm
\textwidth=15cm
%\headsep=.6cm
%\flushbottom
\oddsidemargin=1.2cm
\evensidemargin=1.2cm
%\voffset =-10pt

%%%%%%%%%%%%%%%%%%%%%%%%%%

\def\R{{\rm I\kern-2ptR}}
\def\N{{\rm I\kern-2ptN}}


\def\emptyset{\mathop{\raisebox{.1ex}{$\not
\mathrel{\raisebox{.1ex}{$\scriptstyle\bigcirc$}}$}}}


\def\qed{\hskip .6em \raise1.8pt\hbox{\vrule height4pt
width6pt depth2pt}}
\def\qedd{\hskip .4em \raise1.8pt\hbox{\vrule height3pt
width5pt depth1.8pt}}
\def\sq{\hskip .6em \raise1.8pt\hbox{\vrule
height4pt width6pt depth2pt}}

\def\ENGLISH{\marg{\underbar{\qed ENGLISH}}}
\def\ENGLISHH{\marg{{!.ENGLISH}}}



\font\head=cmr10 at 12 pt
\font\titlos=cmb10 at 18 pt
\font\titloss=cmb10 at 16 pt


\def\leaderfill{\leaders\hbox to 1em{\hss.\hss}\hfill}

%\newtheorem{remark}{Remark}





%%4.11%% \def\pitchfork{{\Psi}}
%%4.11%% \def\mathbf{\bf }
%%4.11%% \def\mathfrak{\it }
\def\mathds{\mathbb}



\font\head=cmr10 at 12 pt \font\titlos=cmb10 at 18 pt
\font\titloss=cmb10 at 16 pt







\begin{document}
%\baselineskip 24 pt

\title[]{Garling sequence spaces}

\author{Ben Wallis}
\address{Department of Mathematical Sciences, Northern Illinois University, DeKalb, IL 60115}
\email{benwallis@live.com}

\begin{abstract}By generalizing a construction of Garling, for each $1\leqslant p<\infty$ and each normalized, nonincreasing sequence of positive numbers $w\in c_0\setminus\ell_1$ we exhibit an $\ell_p$-saturated, complementably homogeneous Banach space $g(w,p)$ related to the Lorentz sequence space $d(w,p)$.  Using methods originally developed for studying $d(w,p)$, we show that $g(w,p)$ admits a unique (up to equivalence) subsymmetric basis, although when $w=(n^{-\theta})_{n=1}^\infty$ for some $0<\theta<1$, it does not admit a symmetric basis.  We then discuss some additional properties of $g(w,p)$ related to uniform convexity and superreflexivity.\end{abstract}


\thanks{The author thanks Greg Martin and B\"unyamin Sar{\i} for their helpful comments.\\
\indent MSC10:  46B25, 46B45, 46B04\\
\indent Keywords:  functional analysis, Banach spaces, Lorentz sequence spaces, subsymmetric bases, symmetric bases}

\maketitle
\theoremstyle{plain}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Introduction


\section{Introduction}

In this paper we shall generalize a Banach space constructed by D.J.H. Garling in \cite{Ga68} in order to define a new family of Banach spaces $g(w,p)$, $1\leq p<\infty$, which can be seen as variations on the family of Lorentz sequence spaces $d(w,p)$.  %Here, $w=(w_n)_{n=1}^\infty\in c_0\setminus\ell_1$ is a normalized ($\|w\|_\infty=w_1=1$), nonincreasing ($w_{n+1}\leq w_n$) sequence of positive real numbers. In Theorem \ref{main} we show that these spaces are uniformly $\ell_p$-saturated in the sense that every infinite-dimensional closed subspace admits, for each $\epsilon>0$, a further subspace which is $(1+\epsilon)$-isomorphic to $\ell_p$.  Furthermore, $g(w,p)$ always admits a unique (up to equivalence) subsymmetric basis (see Theorem \ref{unique-subsymmetric}), although in case $w=(n^{-\theta})_{n=1}^\infty$, $0<\theta<1$, it fails to admit any symmetric basis (see Example \ref{not-symmetric}).
%The norm for sequences in $g(w,p)$ is defined in almost the same way as that of the Lorentz sequence spaces $d(w,p)$, only replacing permutations of sequences with subsequences.
The original motivation behind Garling's construction was to exhibit a subsymmetric basis which is not symmetric, and his original construction in \cite{Ga68} consisted of $g(w,1)$ with $w=(n^{-1/2})_{n=1}^\infty$.  Similar modifications have appeared in past literature, namely the spaces constructed in \cite{Pu76} and \cite[\S5]{DOSZ11}.  The one in \cite{Pu76} is a modification of $g(w,2)$ for $w=(n^{-1/2})_{n=1}^\infty$, and forms a uniformly convex space whose canonical basis is subsymmetric but not symmetric.  The space from \cite[\S5]{DOSZ11} can be viewed as being constructed from variations of $g(w,1)$ for various choices of weights $w$, and yields a canonical basis which is 1-greedy and subsymmetric but not symmetric.

Both the new Garling sequence spaces $g(w,p)$ and the older Lorentz sequence spaces $d(w,p)$ are defined by taking the completion of $c_{00}$ under norms $\|\cdot\|_g$ or $\|\cdot\|_d$, respectively.  The only difference between these norms is that $\|\cdot\|_g$ is defined by taking a certain supremum over subsequences instead of permutations of sequences, as is the case for $\|\cdot\|_d$.  Consequently, many results which have been established in previous literature for Lorentz sequence spaces carry over, with more or less the same proofs, for the Garling sequence spaces $g(w,p)$.  Indeed, most of our results below are just minor adaptations of results in \cite{ACL73}, \cite{CL74}, and \cite{Al75}.
%In \cite{ACL73}, \cite{CL74}, and \cite{Al75}, the Lorentz sequence spaces $d(w,p)$ were extensively studied, and the methods in those three papers shall form the backbone for our results here.  Indeed, many of the properties of $d(w,p)$ carry over in some fashion to $g(w,p)$, including the fact that they are always complementably $\ell_p$-saturated and admit a unique (up to equivalence) subsymmetric basis.  However, for certain choices of $w$ (see Example \ref{not-symmetric} below), the Garling sequence space $g(w,p)$ fails to admit a symmetric basis.


Let us recall some basic definitions and notation which we will need later.  Denote by $c_{00}$ the space of all scalar sequences (i.e., elements of $\mathbb{R}^\mathbb{N}$ or of $\mathbb{C}^\mathbb{N}$) with finitely many nonzero entries.  If $(x_n)_{n=1}^\infty$ is a basis for a Banach space $X$, and $x=\sum_{n=1}^\infty a_nx_n$, then we denote $\text{supp }x:=\{n\in\mathbb{N}:a_n\neq 0\}$, i.e. the set of indices corresponding to the nonzero entries of $x$ (which is called the {\it support} of $x$).  When the basis $(x_n)_{n=1}^\infty$ is clear from context, we may write $x=(a_n)_{n=1}^\infty$ in place of $x=\sum_{n=1}^\infty a_nx_n$.

Let $X$ and $Y$ denote infinite-dimensional Banach spaces.  The space $X$ is said to be {\bf $\boldsymbol{Y}$-saturated} just in case every infinite-dimensional closed subspace of $X$ admits a further subspace $Z$ which is isomorphic to $Y$, written $Z\approx Y$.
%; if furthermore we can always choose $Z$ to be complemented in $X$ then we say that $X$ is {\bf complementably $\boldsymbol{Y}$-saturated}.
(In the literature, beging $Y$-saturated is sometimes called being {\it hereditarily} $Y$.)  Now suppose $(x_n)_{n=1}^\infty$ and $(y_n)_{n=1}^\infty$ are sequences in $X$ and $Y$, respectively.  If there exists $C\in[1,\infty)$ such that
\[\norm{\sum_{n=1}^\infty a_nx_n}_X\leq C\norm{\sum_{n=1}^\infty a_ny_n}_Y\]
for all finitely-supported sequences of scalars $(a_n)_{n=1}^\infty\in c_{00}$, then we say that $(y_n)_{n=1}^\infty$ {\bf $\boldsymbol{C}$-dominates} $(x_n)_{n=1}^\infty$, and write $(x_n)_{n=1}^\infty\lesssim_C(y_n)_{n=1}^\infty$.  In case $C$ does not matter, we will simply say that $(y_n)_{n=1}^\infty$ {\bf dominates} $(x_n)_{n=1}^\infty$, and write $(x_n)_{n=1}^\infty\lesssim(y_n)_{n=1}^\infty$.  Whenever $(x_n)_{n=1}^\infty\lesssim_C(y_n)_{n=1}^\infty$ and $(y_n)_{n=1}^\infty\lesssim_C(x_n)_{n=1}^\infty$, we say that $(x_n)_{n=1}^\infty$ and $(y_n)_{n=1}^\infty$ are {\bf $\boldsymbol{C}$-equivalent}, and write $(x_n)_{n=1}^\infty\approx_C(y_n)_{n=1}^\infty$ or, simply, $(x_n)_{n=1}^\infty\approx(y_n)_{n=1}^\infty$.

A basic sequence $(x_n)_{n=1}^\infty$ is called {\bf subsymmetric} just in case it is unconditional and equivalent to each of its subsequences.  It is called {\bf symmetric} whenever it is unconditional and equivalent to each of its permutations.  By a uniform boundedness argument (cf., e.g., \cite[\S3.a]{LT77} for details), it follows that if $(x_n)_{n=1}^\infty$ is a subsymmetric basic sequence then there is a uniform constant $C\geq 1$ such that if $(x_{n_k})_{k=1}^\infty$ is any subsequence and $(\epsilon_k)_{k=1}^\infty$ is any choice of signs (or, in the complex case, elements of the complex unit circle $\mathbb{T}$), then $(x_n)_{n=1}^\infty$ is $C$-equivalent to $(\epsilon_kx_{n_k})_{k=1}^\infty$.  In this case we say that $(x_n)_{n=1}^\infty$ is {\bf $\boldsymbol{C}$-subsymmetric}.  Similarly, if $(x_n)_{n=1}^\infty$ is symmetric then there is $C\geq 1$ such that $(x_n)_{n=1}^\infty$ is $C$-equivalent to each $(\epsilon_nx_{\sigma(n)})_{n=1}^\infty$, where $\sigma$ is a permutation of $\mathbb{N}$, in which case we say that $(x_n)_{n=1}^\infty$ is {\bf $\boldsymbol{C}$-symmetric}.  Note that $C$-symmetry implies $C$-subsymmetry, which in turn implies $C$-unconditionality.


Most other notation is standard, such as might appear, for instance, in \cite{LT77} or \cite{AK06}.  However, we still need to introduce notation for the spaces under consideration in this paper.






\subsection{Lorentz sequence spaces}

We denote by
\[\textbf{W}:=\left\{(w_n)_{n=1}^\infty\in c_0\setminus\ell_1:1=w_1\geq w_2\geq w_3\geq\cdots>0\right\}\]
the set of all normalized null sequences of positive nonincreasing weights which are not summable.  We also let $\Pi$ denote the set of permutations on $\mathbb{N}$.  Then for any $1\leq p<\infty$ and any $w\in\textbf{W}$, we define a {\bf Lorentz sequence space} $d(w,p)$ as the completion of $c_{00}$ under the norm $\norm{\cdot}_d$ defined by the rule
\[\norm{(a_n)_{n=1}^\infty}_d=\sup_{\sigma\in\Pi}\norm{(a_{\sigma(n)}w_n^{1/p})_{n=1}^\infty}_p,\;\;\;(a_n)_{n=1}^\infty\in c_{00}.\]
Here, $\|\cdot\|_p$ denotes the $\ell_p$-norm defined by
$$
\|(a_n)_{n=1}^\infty\|_p=\left(\sum_{n=1}^\infty|a_n|^p\right)^{1/p},\;\;\;(a_n)_{n=1}^\infty\in\ell_p.
$$
We will usually denote by $(d_n)_{n=1}^\infty$ the unit vector basis of $d(w,p)$.  Observe that the choice of letter ``d'' for the Lorentz sequence spaces follows standard conventions in the literature.



\subsection{Garling sequence spaces}

Let $\mathbb{N}^\uparrow$ denote the set of all strictly increasing sequences of positive integers, i.e. the set of all sequences
\[(n_1<n_2<n_3<\cdots)\in\mathbb{N}^\mathbb{N}.\]
Then for any $1\leq p<\infty$ and any $w\in\textbf{W}$, we define a {\bf Garling sequence space} $g(w,p)$ as the completion of $c_{00}$ under the norm $\norm{\cdot}_g$ defined by the rule
\[\norm{(a_n)_{n=1}^\infty}_g=\sup_{(n_k)_{k=1}^\infty\in\mathbb{N}^\uparrow}\norm{(a_{n_k}w_k^{1/p})_{k=1}^\infty}_p,\;\;\;(a_n)_{n=1}^\infty\in c_{00}.\]
We will usually denote by $(g_n)_{n=1}^\infty$ the unit vector basis of $g(w,p)$.



We might want to also say a few words about the restrictions on $w$ in the definition of $g(w,p)$.  The choice of $w_1=1$ gives us normalization, and requiring that $w=(w_n)_{n=1}^\infty$ be nonincreasing leads to relatively simple norm computations.  The further conditions $w\in c_0$ and $w\notin\ell_1$ prevent us, respectively, from having $g(w,p)\approx\ell_p$ or $g(w,p)\approx c_0$.



We shall need two basic results on subsymmetric bases.  We omit the proofs since they are practically identical to the analogous results for symmetric basic sequences in \cite{ACL73}.



\begin{proposition}[see {\cite[Proposition 3]{ACL73}}]\label{subsymmetric-support-equivalent}Let $X$ be a Banach space with a subsymmetric basis $(x_n)_{n=1}^\infty$ and let $(y_i)_{i=1}^\infty$ be a seminormalized block basic sequence in of $(x_n)_{n=1}^\infty$.  If there is $L\in\mathbb{N}$ such that $\#\text{supp }y_i\leq L$ for all $i\in\mathbb{N}$, then $(y_i)_{i=1}^\infty$ is equivalent to $(x_n)_{n=1}^\infty$.\end{proposition}




\begin{proposition}[see {\cite[Proposition 4]{ACL73}}]\label{subsymmetric-dominates}Let $X$ be a Banach space with a subsymmetric basis $(x_n)_{n=1}^\infty$, and let
\[y_n=\sum_{i=p_n}^{p_{n+1}-1}a_ix_i,\;\;\;n\in\mathbb{N},\]
form a block basic sequence satisfying
\begin{equation}\label{4}\inf_{n\in\mathbb{N}}\sup_{p_n\leq i\leq p_{n+1}-1}|a_i|>0.\end{equation}
Then $(y_n)_{n=1}^\infty\gtrsim(x_n)_{n=1}^\infty$.\end{proposition}


We also record the following fact, which is well-known and has an elementary proof which we omit.  %(It can also be deduced from Rosenthal's $\ell_1$ Theorem together with Proposition \ref{subsymmetric-support-equivalent} above.)

\begin{proposition}\label{subsymmetric-weakly-null}Every subsymmetric basic sequence in a Banach space is either weakly null or else equivalent to the canonical basis of $\ell_1$.\end{proposition}

This paper is organized as follows.  In section 2 we collect some basic facts about the spaces $g(w,p)$.  Section 3 is where we show that $g(w,p)$ always admits a unique subsymmetric basis but, for certain $w\in\textbf{W}$, fails to admit a symmetric basis.  Finally, in section 4, we prove that $g(w,p)$ is never uniformly convex, and that in some cases it is reflexive but not superreflexive.  %We stress once more that the methods in \cite{ACL73}, \cite{CL74}, and \cite{Al75}, where the Lorentz sequence spaces $d(w,p)$ were extensively studied, shall form the backbone for our results here.



\section{Basic results on $g(w,p)$}



In this section, we shall collect some basic facts about $g(w,p)$, ultimately proving the following result.


\begin{theorem}\label{main}Let $1\leq p<\infty$ and $w=(w_n)_{n=1}^\infty\in\emph{\textbf{W}}$, and denote by $(g_n)_{n=1}^\infty$ the canonical basis for $g(w,p)$.
\begin{itemize}
\item[(i)]  $(g_n)_{n=1}^\infty$ is weakly null.
\item[(ii)]  Every basic sequence in $g(w,p)$ equivalent to $(g_n)_{n=1}^\infty$ admits a subsequence spanning a subspace complemented in $g(w,p)$.
\item[(iii)]  If $1<p<\infty$ then $g(w,p)$ is reflexive, but if $p=1$ then $g(w,1)$ is nonreflexive.
\item[(iv)]  No subspace of $\ell_p$ is isomorphic to $g(w,p)$.
\item[(v)]  For every $\epsilon>0$ and every infinite-dimensional closed subspace $Y$ of $g(w,p)$ there exists a further subspace $Z\subseteq Y$ which is $(1+\epsilon)$-isomorphic to $\ell_p$.
%In case $p=1$, this subspace $Z$ can be chosen to be $(1+\epsilon)$-complemented in $g(w,1)$.
\item[(vi)]  In case $p=1$, for every infinite-dimensional closed subspace $Y$ of $g(w,1)$, there exists a further subspace $Z\subseteq Y$ which is isomorphic to $\ell_1$ and complemented in $g(w,p)$.
\item[(vii)]  For every $\epsilon>0$, the space $g(w,p)$ contains a closed subspace which is $(1+\epsilon)$-isomorphic to $\ell_p$ and $(1+\epsilon)$-complemented in $g(w,p)$.
\end{itemize}\end{theorem}

\begin{remark}We say that a Banach space $X$ is {\bf complementably homogeneous} whenever for every closed subspace $Y$ of $X$ such that $Y\approx X$, there exists another closed subspace $Z$ such that $Z$ is complemented in $X$, $Z\approx X$, and $Z\subseteq Y$.  Observe that condition (ii) in Theorem \ref{main} above, together with subsymmetry, implies that $g(w,p)$ is complementably homogeneous.\end{remark}


Let us begin with the following.



\begin{proposition}\label{lp-dominates}Let $1\leq p<\infty$ and $w=(w_n)_{n=1}^\infty\in\emph{\textbf{W}}$, and let $(g_n)_{n=1}^\infty$ denote the canonical basis of $g(w,p)$.  Then any normalized block basic sequence of $(g_n)_{n=1}^\infty$ is 1-dominated by the canonical basis for $\ell_p$.\end{proposition}

\begin{proof}Our proof shall follow that of \cite[Proposition 5]{ACL73}.  Let
$$y_n=\sum_{i=p_n}^{p_{n+1}-1}a_ig_i,\;\;\;n\in\mathbb{N},$$
form a normalized block basic sequence in $g(w,p)$, and let $(b_n)_{n=1}^N$, $N\in\mathbb{N}$, be any finite sequence of scalars.  Then we can find a sequence sets $A_1<A_2<\cdots<A_N$ such that $A_n\subseteq\{p_n,p_n+1,\cdots,p_{n+1}-1\}$ and
\[\norm{\sum_{n=1}^Nb_ny_n}_g^p=\sum_{n=1}^N\sum_{i\in A_n}|b_na_i|^pw_{m(i)},\]
where $m(i)=\#\{j\in A_1\cup\cdots\cup A_N:j\leq i\}$.  Notice also that for each $n=1,\cdots,N$, we have
\[\sum_{i\in A_n}|a_i|^pw_{m(i)}\leq\norm{y_n}_g^p=1\]
so that
\[\norm{\sum_{n=1}^Nb_ny_n}_g^p=\sum_{n=1}^N\sum_{i\in A_n}|b_na_i|^pw_{m(i)}=\sum_{n=1}^N|b_n|^p\sum_{i\in A_n}|a_i|^pw_{m(i)}\leq\sum_{n=1}^N|b_n|^p.\]
It follows that $(y_n)_{n=1}^\infty$ is 1-dominated by the canonical basis for $\ell_p$.\end{proof}



When characterizing the norm of $x=\sum_{n=1}^\infty a_nd_n\in d(w,p)$, $\#\text{supp }x<\infty$, we can always assume $(a_n)_{n=1}^\infty$ is positive and nonincreasing.  However, for $g(w,p)$ this is not always possible.  In particular, we will need a substitute method to deal with block bases in $g(w,p)$, which we give below.


\begin{proposition}\label{1-dominated-blocks}Let $1\leq p<\infty$ and $w=(w_n)_{n=1}^\infty\in\emph{\textbf{W}}$, and let $(g_n)_{n=1}^\infty$ denote the canonical basis of $g(w,p)$.  Suppose
\[y_n=\sum_{i=p_n}^{p_{n+1}-1}a_ig_i,\;\;\;n\in\mathbb{N}\]
forms a block basic sequence.  Then there exists another block basis formed by vectors
\[\hat{y}_n=\sum_{\ell=\hat{p}_n}^{\hat{p}_{n+1}-1}a_{i_\ell}g_\ell,\;\;\;n\in\mathbb{N},\]
such that
\begin{itemize}\item[(i)] $a_{i_\ell}\neq 0$ for any $\ell\in\mathbb{N}$;
\item[(ii)] $\hat{p}_1=1$;
\item[(iii)] $(i_\ell)_{\ell=\hat{p}_n}^{\hat{p}_{n+1}-1}$ is a subsequence of $(i)_{i=p_n}^{p_{n+1}-1}$ for each $n\in\mathbb{N}$;
\item[(iv)] $\displaystyle\norm{y_n}_g=\norm{\hat{y}_n}_g=\left(\sum_{\ell=\hat{p}_n}^{\hat{p}_{n+1}-1}|a_{i_\ell}|^pw_{\ell-\hat{p}_n+1}\right)^{1/p}$ for each $n\in\mathbb{N}$; and
\item[(v)] $(\hat{y}_n)_{n=1}^\infty$ is 1-dominated by $(y_n)_{n=1}^\infty$.\end{itemize}\end{proposition}

\begin{proof}Notice that for each $n\in\mathbb{N}$ we can find $A_n\subseteq\{p_n,\cdots,p_{n+1}-1\}$ such that if $m(i)=\#\{r\in A_n:r\leq i\}$ for each $p_n\leq i\leq p_{n+1}-1$ then
\[\norm{y_n}_{g}=\left(\sum_{i\in A_n}|a_i|^pw_{m(i)}\right)^{1/p}.\]
Without loss of generality we may assume that $a_i\neq 0$ whenever $i\in A_n$ for some $n\in\mathbb{N}$.  We can now define, for each $p_n\leq i\leq p_{n+1}-1$,
$$\tilde{a}_i=\left\{\begin{array}{ll}a_i&\text{ if }i\in A_n,\text{ and}\\0&\text{ otherwise.}\end{array}\right.$$
Now set, for each $n\in\mathbb{N}$,
\[\tilde{y}_n=\sum_{i=p_n}^{p_{n+1}}\tilde{a}_ig_i.\]
Then $(\tilde{y}_n)_{n=1}^\infty$ is a normalized block basic sequence which, by 1-unconditionality, is 1-dominated by $(y_n)_{n=1}^\infty$.

Next we shall construct the block basic sequence $(\hat{y}_n)_{n=1}^\infty$ as above, by defining the sequences $(a_{i_\ell})_{\ell=1}^\infty$ and $(\hat{p}_n)_{n=1}^\infty$.  Let $(\tilde{a}_{i_\ell})_{\ell=1}^\infty$ be the subsequence which consists precisely of the nonzero values of $(\tilde{a}_i)_{i=1}^\infty$.  Immediately, (i) holds.
%Notice that $\tilde{a}_{i_\ell}=a_{i_\ell}$ for each $\ell\in\mathbb{N}$, so that (i) holds.
It is also clear that there exist $1=\hat{p}_1<\hat{p}_2<\hat{p}_3<\cdots\in\mathbb{N}$ so that (ii) and (iii) also hold.  Now, observe that $\norm{y_n}_{g}=\norm{\tilde{y}_n}_{g}=\norm{\hat{y}_n}_{g}$ which gives us (iv).  Finally, we have
\[(\hat{y}_n)_{n=1}^\infty\approx_1(\tilde{y}_n)_{n=1}^\infty\lesssim_1(y_n)_{n=1}^\infty\]
to get (v).\end{proof}

Now we are able to prove the following powerful result, an adaptation of \cite[Lemma 1]{ACL73} and \cite[Lemma 15]{CL74}.


\begin{theorem}\label{subsequence-equivalent-lp}Let $1\leq p<\infty$ and $w=(w_n)_{n=1}^\infty\in\emph{\textbf{W}}$, and let $(g_n)_{n=1}^\infty$ denote the canonical basis of $g(w,p)$.  Suppose
\[y_n=\sum_{i=p_n}^{p_{n+1}-1}a_ig_i\]
forms a normalized block basic sequence, and satisfies
\[\lim_{i\to\infty}a_i=0.\]
Let $(\hat{y}_n)_{n=1}^\infty$ be the sequence from Proposition \ref{1-dominated-blocks} corresponding to $(y_n)_{n=1}^\infty$.  Then for each $\epsilon>0$ there exists a subsequence $(y_{n_k})_{k=1}^\infty$ which is $(1+\epsilon)$-equivalent to the canonical basis $(f_n)_{n=1}^\infty$ of $\ell_p$, and such that $[\hat{y}_{n_k}]_{k=1}^\infty$ is $(1+\epsilon)$-complemented in $g(w,p)$.\end{theorem}

\begin{remark}The $d(w,p)$-analog for Theorem \ref{subsequence-equivalent-lp} given in \cite[Lemma 1]{ACL73} and \cite[Lemma 15]{CL74} yields a stronger conclusion in the sense that every normalized block sequence in $d(w,p)$ with coefficients tending to zero admits a subsequence which is $(1+\epsilon)$-equivalent to $\ell_p$ and $(1+\epsilon)$-complemented in $d(w,p)$.  Consequently, $d(w,p)$ is uniformly subprojective.  Because the analogous result for $g(w,p)$ requires passing to $(\hat{y}_n)_{n=1}^\infty$, subprojectivity does not follow in the same way that it does for $d(w,p)$.  An alternative method for showing subprojectivity of $g(w,1)$ (i.e., in the special case $p=1$) is given in the proof of Theorem \ref{main}(vi).  However, the uniform constant is not established there; moreover, it remains an open question whether $g(w,p)$ is subprojective for $1<p<\infty$.  See \cite{OS15} for an overview of basic facts on subprojectivity.\end{remark}

\begin{proof}[Proof of Theorem \ref{subsequence-equivalent-lp}]The first part of our proof follows that of \cite[Lemma 1]{ACL73}.  Let
\[\hat{y}_n=\sum_{\ell=\hat{p}_n}^{\hat{p}_{n+1}-1}a_{i_\ell}g_\ell,\;\;\;n\in\mathbb{N},\]
form a normalized block basic sequence as in Proposition \ref{1-dominated-blocks}, and fix any $\epsilon>0$.  Let $\phi\in(0,1)$ be such that $\phi^{-1/p}\leq\phi^{-1}<1+\epsilon$.

We claim that there exists a block basic sequence of the form
\[z_n=\sum_{\ell=q_n}^{q_{n+1}-1}b_\ell g_\ell,\;\;\;n\in\mathbb{N},\]
and a subsequence $(\hat{y}_{n_k})_{k=1}^\infty$, such that the following two properties are satisfied for every $k\in\mathbb{N}$.
\begin{itemize}\item[($*$)]  the coefficients of $z_k$ are the same as the coefficients of $\hat{y}_{n_k}$, that is, $q_{k+1}-q_k=\hat{p}_{n_k+1}-\hat{p}_{n_k}$ and $b_{q_k+\ell}=a_{i_{\hat{p}_{n_k}+\ell}}$ for $\ell=0,\cdots q_{k+1}-q_k-1$; and
\item[($**$)]  $\sum_{\ell=q_k}^{q_{k+1}-1}|b_\ell|^pw_\ell\geq\phi$.\end{itemize}

Let us begin by setting $q_1=1$, and then let us suppose that we have defined $q_k$ for some $k\in\mathbb{N}$.  Due to $w_n\to 0$, we can find $j\in\mathbb{N}$ such that
\[\sum_{\ell=j+1}^{j+q_k}w_\ell<\frac{1-\phi}{2}.\]
Note that since $a_{i_\ell}\to 0$ we must have $\hat{p}_{n+1}-\hat{p}_n\to\infty$.  Thus, we may select $h\in\mathbb{N}$ such that $\hat{p}_{h+1}-\hat{p}_h>j+q_k$ and $|a_{i_\ell}|<\frac{1-\phi}{2j}$ for all $\ell\geq\hat{p}_h$.  Set $q_{k+1}=\hat{p}_{h+1}-\hat{p}_h+q_k$ and $b_{q_k+\ell}=a_{i_{\hat{p}_h+\ell}}$ for all $\ell=0,\cdots,q_{k+1}-q_k-1$.  This way, setting $\hat{y}_{n_k}=\hat{y}_h$ immediately satisfies ($*$).  Recall from property (iv) in Proposition \ref{1-dominated-blocks} that
\[1=\norm{\hat{y}_h}_{g}^p=\sum_{\ell=\hat{p}_h}^{\hat{p}_{h+1}-1}|a_{i_\ell}|^pw_{\ell-\hat{p}_h+1},\]
and therefore that
\begin{multline}\label{1}\sum_{\ell=q_k}^{q_{k+1}-1}|b_\ell|^pw_\ell
=\sum_{\ell=\hat{p}_h}^{\hat{p}_{h+1}-1}|a_{i_\ell}|^pw_{\ell+q_k-\hat{p}_h}
\\=\sum_{\ell=\hat{p}_h}^{\hat{p}_{h+1}-1}|a_{i_\ell}|^pw_{\ell-\hat{p}_h+1}-\sum_{\ell=\hat{p}_h}^{\hat{p}_{h+1}-1}|\hat{a}_\ell|^p(w_{\ell-\hat{p}_h+1}-w_{\ell+q_k-\hat{p}_h})
\\=1-\sum_{\ell=\hat{p}_h}^{\hat{p}_h+j-1}|a_{i_\ell}|^p(w_{\ell-\hat{p}_h+1}-w_{\ell+q_k-\hat{p}_h})-\sum_{\ell=\hat{p}_h+j}^{\hat{p}_{h+1}-1}|a_{i_\ell}|^p(w_{\ell-\hat{p}_h+1}-w_{\ell+q_k-\hat{p}_h}).\end{multline}
Note that since $(w_\ell)_{\ell=1}^\infty$ is nonincreasing with $0<w_\ell\leq 1$, $\ell\in\mathbb{N}$, and $q_k\geq 1$, we have
\[0\leq w_{\ell-\hat{p}_h+1}-w_{\ell+q_k-\hat{p}_h}<1,\;\;\;\ell\in\mathbb{N}.\]
Together with the facts that there are $j$ values between $\hat{p}_h$ and $\hat{p}_h+j-1$, as well as $|a_{i_\ell}|^p<\frac{1-\phi}{2j}$ for all $\ell\geq\hat{p}_h$, we now have
\begin{equation}\label{2}
\sum_{\ell=\hat{p}_h}^{\hat{p}_h+j-1}|a_{i_\ell}|^p(w_{\ell-\hat{p}_h+1}-w_{\ell+q_k-\hat{p}_h})
<\sum_{\ell=\hat{p}_h}^{\hat{p}_h+j-1}|a_{i_\ell}|^p
<\frac{1-\phi}{2j}\sum_{\ell=\hat{p}_h}^{\hat{p}_h+j-1}1=\frac{1-\phi}{2}.
\end{equation}
Also, note that, due again to $|a_{i_\ell}|^p<\frac{1-\phi}{2j}<1$ for all $\ell\geq\hat{p}_h$, as well as the facts $\sum_{\ell=j+1}^{j+q_k}w_\ell<\frac{1-\phi}{2}$ and $\hat{p}_{h+1}+q_k-2\geq\hat{p}_{h+1}-1$, we have
\begin{multline}\label{3}
\sum_{\ell=\hat{p}_h+j}^{\hat{p}_{h+1}-1}|a_{i_\ell}|^p(w_{\ell-\hat{p}_h+1}-w_{\ell+q_k-\hat{p}_h})
<\sum_{\ell=\hat{p}_h+j}^{\hat{p}_{h+1}-1}(w_{\ell-\hat{p}_h+1}-w_{\ell+q_k-\hat{p}_h})
\\=\sum_{\ell=\hat{p}_h+j}^{\hat{p}_{h+1}-1}w_{\ell-\hat{p}_h+1}-\sum_{\ell=\hat{p}_h+j+q_k-1}^{\hat{p}_{h+1}+q_k-2}w_{\ell-\hat{p}_h+1}
\\<\sum_{\ell=\hat{p}_h+j}^{\hat{p}_h+j+q_k-1}w_{\ell-\hat{p}_h+1}
=\sum_{\ell=j+1}^{j+q_k}w_\ell<\frac{1-\phi}{2}.
\end{multline}
Combining \eqref{1}, \eqref{2}, and \eqref{3} gives us ($**$).  This completes the proof of our claim.

Let us show that our new block basic sequence $(z_n)_{n=1}^\infty$ dominates the canonical basis $(f_n)_{n=1}^\infty$ for $\ell_p$.  Let $(c_n)_{n=1}^N$, $N\in\mathbb{N}$, be any finite sequence of scalars.  Then
\[\phi\sum_{n=1}^N|c_n|^p\leq\sum_{n=1}^N|c_n|^p\sum_{\ell=q_n}^{q_{n+1}-1}|b_\ell|^pw_\ell=\sum_{n=1}^N\sum_{\ell=q_n}^{q_{n+1}-1}|c_nb_\ell|^pw_\ell\leq\norm{\sum_{n=1}^Nc_nz_n}_{g}^p\]
so that 
\[\left(\sum_{n=1}^N|c_n|^p\right)^{1/p}\leq\phi^{-1/p}\norm{\sum_{n=1}^Nc_nz_n}_{g}.\]
Hence, $(f_n)_{n=1}^\infty$ is $\phi^{-1/p}$-dominated by $(z_n)_{n=1}^\infty$.  On the other hand, $(z_n)_{n=1}^\infty$ is 1-equivalent to $(\hat{y}_{n_k})_{k=1}^\infty$ by ($*$) together with 1-subsymmetry of $(g_n)_{n=1}^\infty$.  Recall that by property (v) in Proposition \ref{1-dominated-blocks}, the sequence $(\hat{y}_n)_{n=1}^\infty$ is 1-dominated by $(y_n)_{n=1}^\infty$, and that by Proposition \ref{lp-dominates} the sequence $(y_{n_k})_{k=1}^\infty$ is 1-dominated by $(f_n)_{n=1}^\infty$.  We now have the diagram
\begin{equation}\label{y-z-equivalent}(f_n)_{n=1}^\infty\lesssim_{\phi^{-1/p}}(z_n)_{n=1}^\infty\approx_1(\hat{y}_{n_k})_{k=1}^\infty\lesssim_1(y_{n_k})_{k=1}^\infty\lesssim_1(f_n)_{n=1}^\infty\end{equation}
so that $(y_{n_k})_{k=1}^\infty$ is $\phi^{-1/p}$-equivalent, and hence also $(1+\epsilon)$-equivalent, to $(f_n)_{n=1}^\infty$.
%
%(ii)
%
%Since $y_n=\hat{y}_n$ for all $n\in\mathbb{N}$, we have
%\begin{equation}\label{assumption-ii}(b_\ell)_{\ell=q_k}^{q_{k+1}-1}=(a_{i_\ell})_{\ell=\hat{p}_{n_k}}^{\hat{p}_{n_k+1}-1}=(a_i)_{i=p_{n_k}}^{p_{n_k+1}-1}\;\;\;\text{ for all }k\in\mathbb{N}.\end{equation}

To show that $[\hat{y}_{n_k}]_{k=1}^\infty$ is $(1+\epsilon)$-complemented in $g(w,p)$, we will follow the proof of \cite[Lemma 15]{CL74}.  Define
\[\tilde{z}_k=\sum_{\ell=q_k}^{q_{k+1}-1}|b_\ell|g_\ell,\;\;\;k\in\mathbb{N}.\]
%Notice that by 1-unconditionality, $(\tilde{z}_k)_{k=1}^\infty$ is a normalized block basic sequence which is 1-equivalent to $(z_k)_{k=1}^\infty$, and hence, by \eqref{assumption-ii}, also 1-equivalent to $(y_{n_k})_{k=1}^\infty$.

Let us set, for each $k\in\mathbb{N}$ and each $q_k\leq i\leq q_{k+1}-1$,
\[Pg_i=\frac{|b_i|^{p-1}w_i}{\sum_{\ell=q_k}^{q_{k+1}-1}|b_\ell|^pw_\ell}\tilde{z}_k.\]
Clearly, $P$ extends to a linear operator acting on $\text{span}(g_i)_{i=1}^\infty$.  Moreover, for each $k\in\mathbb{N}$ we have
\[P\tilde{z}_k=P\sum_{i=q_k}^{q_{k+1}-1}|b_i|g_i=\sum_{i=q_k}^{q_{k+1}-1}|b_i|\frac{|b_i|^{p-1}w_i}{\sum_{\ell=q_k}^{q_{k+1}-1}|b_\ell|^pw_\ell}\tilde{z}_k=\tilde{z}_k\]
so that $P$ is a projection onto $\text{span}(\tilde{z}_k)_{k=1}^\infty$.  Since $(\tilde{z}_k)_{k=1}^\infty$ is a normalized block sequence, it is 1-dominated by $(f_k)_{k=1}^\infty$ by Proposition \ref{lp-dominates}.  Since also ($**$) holds, this gives us, for each $(c_i)_{i=1}^\infty\in c_{00}$,
\begin{multline*}\norm{P\sum_{i=1}^\infty c_ig_i}^p=\norm{P\sum_{k=1}^\infty\sum_{i=q_k}^{q_{k+1}-1}c_ig_i}^p
=\norm{\sum_{k=1}^\infty\left(\sum_{i=q_k}^{q_{k+1}-1}\frac{c_i|b_i|^{p-1}w_i}{\sum_{\ell=q_k}^{q_{k+1}-1}|b_\ell|^pw_\ell}\right)\tilde{z}_k}^p
\\\leq\sum_{k=1}^\infty\left|\sum_{i=q_k}^{q_{k+1}-1}\frac{c_i|b_i|^{p-1}w_i}{\sum_{\ell=q_k}^{q_{k+1}-1}|b_\ell|^pw_\ell}\right|^p
\leq\phi^{-p}\sum_{k=1}^\infty\left|\sum_{i=q_k}^{q_{k+1}-1}c_i|b_i|^{p-1}w_i\right|^p\end{multline*}
Then, using H\"older's inequality with with $\frac{p}{p-1}$ conjugate to $p$, and viewing
$$c_i|b_i|^{p-1}w_i=(c_iw_i^{1/p})\cdot(|b_i|^{p-1}w_i^{(p-1)/p}),$$
we get, together with $\sum_{i=q_k}^{q_{k+1}-1}|b_i|^pw_i\leq\|z_k\|^p=1$,
\begin{multline*}\phi^{-p}\sum_{k=1}^\infty\left|\sum_{i=q_k}^{q_{k+1}-1}c_i|b_i|^{p-1}w_i\right|^p\leq\phi^{-p}\sum_{k=1}^\infty\left(\sum_{i=q_k}^{q_{k+1}-1}|c_i|^pw_i\right)\left(\sum_{i=q_k}^{q_{k+1}-1}|b_i|^pw_i\right)^{p-1}\\\leq\phi^{-p}\sum_{k=1}^\infty\sum_{i=q_k}^{q_{k+1}-1}|c_i|^pw_i\leq\phi^{-p}\norm{\sum_{i=1}^\infty c_ig_i}_{g}^p.\end{multline*}
It follows that $P$ is bounded by $\phi^{-1}$, and hence has a $\phi^{-1}$-bounded continuous extension $\widetilde{P}\in\mathcal{L}(g(w,p))$ such that $\widetilde{P}$ is a projection onto $[\tilde{z}_k]_{k=1}^\infty$.

Let us define operators $\Phi_1,\Theta_1,\Phi_2,\Theta_2\in\mathcal{L}(g(w,p))$ satisfying
\begin{equation}\label{desired-chains}\hat{y}_{n_k}\stackrel{\Phi_1}{\mapsto}z_k\stackrel{\Phi_2}{\mapsto}\tilde{z}_k\;\;\;\text{ and }\;\;\;\tilde{z}_k\stackrel{\Theta_2}{\mapsto}z_k\stackrel{\Theta_1}{\mapsto}\hat{y}_{n_k}\end{equation}
for all $k\in\mathbb{N}$.  We do this by first defining $\Phi_1g_{i_{\hat{p}_{n_k}}+\ell}=g_{q_k+\ell}$ and $\Theta_1g_{q_k+\ell}=g_{i_{\hat{p}_{n_k}}+\ell}$ for all $0\leq\ell\leq q_{k+1}-q_k-1$ and all $k\in\mathbb{N}$, and $\Phi_1 g_i=0$ for all other $i\in\mathbb{N}$.  This makes $\Phi_1,\Theta_1\in\mathcal{L}(g(w,p))$ with $\Phi_1\Theta_1=I_g$, the identity operator on $g(w,p)$, and $\Theta_1\Phi_1$ is the projection onto the space consisting of the closed linear span of vectors $g_{i_{\hat{p}_{n_k}+\ell}}$, $k\in\mathbb{N}$, $0\leq\ell\leq q_{k+1}-q_k-1$.  In particular, $\Theta_1\Phi_1\hat{y}_{n_k}=\hat{y}_{n_k}$ for all $k\in\mathbb{N}$.  Next, set
\[\Phi_2g_i=\frac{|b_i|}{b_i}g_i\;\;\;\text{ and }\;\;\;\Theta_2g_i=\frac{b_i}{|b_i|}g_i,\;\;\;i\in\mathbb{N},\]
which is possible since $b_i\neq 0$ for all $i\in\mathbb{N}$.  Note that $\Phi_2\Theta_2=\Theta_2\Phi_2=I_g$.

By 1-unconditionality of $(g_n)_{n=1}^\infty$, these are all norm-1 operators, and they clearly satisfy the desired chains of mappings in \eqref{desired-chains}.  Let us now form the composition operator
\[\Theta_1\Theta_2\widetilde{P}\Phi_2\Phi_1\in\mathcal{L}(g(w,p)).\]
It is clear from construction that this is a projection onto $[\hat{y}_{n_k}]_{k=1}^\infty$ of norm $\leq\theta^{-1}<1+\epsilon$.
\end{proof}



\begin{corollary}\label{dominates-or-equivalent}Let $1\leq p<\infty$ and $w=(w_n)_{n=1}^\infty\in\emph{\textbf{W}}$, and let $(g_n)_{n=1}^\infty$ denote the canonical basis of $g(w,p)$.  Every normalized block basic sequence of $(g_n)_{n=1}^\infty$ admits a subsequence which dominates $(g_n)_{n=1}^\infty$.\end{corollary}

\begin{proof}This is an immediate consequence of Proposition \ref{subsymmetric-dominates} and Theorem \ref{subsequence-equivalent-lp}, together with the fact that $(g_n)_{n=1}^\infty$ is dominated by the canonical basis for $\ell_p$.\end{proof}

We need just two more preliminary results before proving Theorem \ref{main}.

\begin{proposition}\label{boundedly-complete}Let $1\leq p<\infty$ and $w=(w_n)_{n=1}^\infty\in\emph{\textbf{W}}$, and let $(g_n)_{n=1}^\infty$ denote the canonical basis of $g(w,p)$.  For any $N\in\mathbb{N}$ and any seminormalized block basic sequence $(y_n)_{n=1}^\infty$ of $(g_n)_{n=1}^\infty$, we have
\[\sup_{N\in\mathbb{N}}\norm{\sum_{n=1}^Ny_n}_{g}=\infty.\]
Furthermore, $(g_n)_{n=1}^\infty$ is boundedly-complete.\end{proposition}

\begin{proof}It is clear from the fact that $w\notin\ell_1$ that
\[\sup_{N\in\mathbb{N}}\norm{\sum_{n=1}^Ng_n}_{g}=\infty.\]
By the Principle of Small Perturbations together with Proposition \ref{subsymmetric-dominates} and Theorem \ref{subsequence-equivalent-lp}, we can find a subsequence $(y_{n_k})_{k=1}^\infty$ which either dominates $(g_n)_{n=1}^\infty$ or else is equivalent to $\ell_p$.  Either way, by 1-subsymmetry of $(g_n)_{n=1}^\infty$ we have
\[\sup_{N\in\mathbb{N}}\norm{\sum_{n=1}^Ny_n}_{g}\geq\sup_{N\in\mathbb{N}}\norm{\sum_{k=1}^Ny_{n_k}}_{g}=\infty.\]
Since every unconditional basis which fails to be boundedly complete admits a block basic sequence equivalent to the canonical basis of $c_0$  (cf., e.g., \cite[Theorem 3.3.2]{AK06}), this completes the proof.\end{proof}



\begin{proposition}\label{block-basis-equivalent-lp}Let $1\leq p<\infty$ and $w=(w_n)_{n=1}^\infty\in\emph{\textbf{W}}$, and let $(g_n)_{n=1}^\infty$ denote the canonical basis of $g(w,p)$.  If $(y_n)_{n=1}^\infty$ is a block basic sequence of $(g_n)_{n=1}^\infty$, then for any $\epsilon>0$ there exists a further block basic sequence $(u_n)_{n=1}^\infty$ of $(y_n)_{n=1}^\infty$ which is $(1+\epsilon)$-equivalent to the canonical basis of $\ell_p$
%, and $(1+\epsilon)$-complemented in $g(w,p)$
.\end{proposition}

\begin{proof}Here we shall follow the proof to \cite[Corollary 3]{ACL73}.  Write
\[y'_n=\frac{y_n}{\norm{y_n}_{g}}\]
so that $(y'_n)_{n=1}^\infty$ is a normalized block basic sequence in $g(w,p)$.  By Proposition \ref{boundedly-complete} together with subsymmetry, we can find an increasing sequence $(p_n)_{n=1}^\infty$ of positive integers such that
\[\lim_{n\to\infty}\norm{\sum_{i=p_n}^{p_{n+1}-1}y'_i}_{g}=\infty.\]
Set
\[z_n=\frac{\sum_{i=p_n}^{p_{n+1}-1}y'_i}{\norm{\sum_{i=p_n}^{p_{n+1}-1}y'_i}_{g}}\]
for each $n\in\mathbb{N}$.  It is clear that the coefficients of $(z_n)_{n=1}^\infty$ tend to zero, and so by Theorem \ref{subsequence-equivalent-lp} we can find a subsequence $(z_{n_k})_{k=1}^\infty$ which is $(1+\epsilon)$-equivalent to the canonical basis of $\ell_p$
%, and $(1+\epsilon)$-complemented in $g(w,p)$
.\end{proof}



\begin{proof}[Proof of Theorem \ref{main}]As usual, $(f_n)_{n=1}^\infty$ denotes the canonical basis of $\ell_p$.


(i)  This is an immediate consequence of Proposition \ref{subsymmetric-weakly-null}.

(ii)  We follow the proof of \cite[Corollary 12]{CL74}.  Let $(y_n)_{n=1}^\infty$ be a basic sequence in $g(w,p)$ which is equivalent to $(g_n)_{n=1}^\infty$.  Since $(g_n)_{n=1}^\infty$ and hence $(y_n)_{n=1}^\infty$ are weakly null, we may pass to a subsequence if necessary and apply the Bessaga-Pelczynski Selection Principle, to truncate the vectors $y_n$ to give us an equivalent seminormalized block basic sequence
\[z_n=(S_{p_{n+1}-1}-S_{p_n-1})y_n=\sum_{i=p_n}^{p_{n+1}-1}a_ig_i,\]
where each $S_k\in\mathcal{L}(g(w,p))$, $k\in\mathbb{N}$, denotes the partial sum projection onto the first $k$ coordinates with respect to the basis $(g_n)_{n=1}^\infty$ (and $S_0=0$ is the zero operator).  By the Principle of Small Perturbations, we can choose this block basis so that $[y_n]_{n=1}^\infty$ is complemented in $g(w,p)$ whenever $[z_n]_{n=1}^\infty$ is.  By Theorem \ref{subsequence-equivalent-lp}, since $(g_n)_{n=1}^\infty$ has no subsequence equivalent to the $\ell_p$ basis, we can find a sequence $(a_{i_n})_{n=1}^\infty$ satisfying $p_n\leq i_n<p_{n+1}$ for each $n\in\mathbb{N}$ and $\inf_{n\in\mathbb{N}}|a_{i_n}|=:c>0$.  Let $K\in[1,\infty)$ denote the equivalence constant between $(z_n)_{n=1}^\infty$ and $(g_n)_{n=1}^\infty$, and observe that for any $(b_n)_{n=1}^\infty\in c_{00}$, by 1-subsymmetry of $(g_n)_{n=1}^\infty$ we have
\[\|\sum_{n=1}^\infty\left(\frac{b_{i_n}}{a_{i_n}}\right)z_n\|_g\leq K\|\sum_{n=1}^\infty\left(\frac{b_{i_n}}{a_{i_n}}\right)g_n\|_g\\\leq\frac{K}{c}\|\sum_{n=1}^\infty b_{i_n}g_n\|_g\leq\frac{K}{c}\|\sum_{n=1}^\infty b_ng_n\|_g.\]
Thus we can define a bounded linear projection $P\in\mathcal{L}(g(w,p))$ onto $[z_n]_{n=1}^\infty$ by the rule\[P\sum_{n=1}^\infty b_ng_n=\sum_{n=1}^\infty\left(\frac{b_{i_n}}{a_{i_n}}\right)z_n.\]

(iii)  Consider first the case where $p\neq 1$.  Let us suppose towards a contradiction that $(g_n)_{n=1}^\infty$ fails to be shrinking.  Then due to unconditionality of $(g_n)_{n=1}^\infty$, there exists a block basic sequence $(y_n)_{n=1}^\infty$ of $(g_n)_{n=1}^\infty$ which is equivalent to the canonical $\ell_1$ basis (cf., e.g., \cite[Theorem 3.3.1]{AK06}).  Now, by Proposition \ref{block-basis-equivalent-lp}, we can find a further block basis equivalent to $\ell_p$.  However, this is impossible if $p\neq 1$, as we are assuming.  Thus, $(g_n)_{n=1}^\infty$ is shrinking, and since it is also boundedly complete by Proposition \ref{boundedly-complete}, and unconditional, it must span a reflexive space by a theorem of James (cf., e.g., \cite[Theorem 3.2.13]{AK06}).  This proves the first part of (iv), and the second part about nonreflexivity in case $p=1$ follows since $g(w,1)$ contains a (nonreflexive) copy of $\ell_1$ by Proposition \ref{block-basis-equivalent-lp}.

(iv)  Recall that every seminormalized weakly null sequence in $\ell_p$ admits a subsequence equivalent to $(f_n)_{n=1}^\infty$ (cf., e.g., \cite[Proposition 2.1.3]{AK06}).  So, if $g(w,p)$ embedded into $\ell_p$ then by this fact together with $(g_n)_{n=1}^\infty$ being weakly null and subsymmetry, $(g_n)_{n=1}^\infty$ would be equivalent to $(f_n)_{n=1}^\infty$, which it clearly is not.

(v)  Fix $\epsilon>0$, and let $Y$ be any infinite-dimensional closed subspace of $g(w,p)$.  By a famous result of Mazur, every infinite-dimensional Banach space contains a basic sequence, and so we can find a normalized basic sequence $(y_n)_{n=1}^\infty$ in $Y$.  In case $p>1$, this basic sequence must be weakly null by the reflexivity of $g(w,p)$.  For the case $p=1$, we can assume by Rosenthal's $\ell_1$ Theorem (cf., e.g., \cite[Theorem 10.2.1]{AK06}) together with James' $\ell_1$ Distortion Theorem (cf., e.g., \cite[Theorem 10.3.1]{AK06}) that $(y_n)_{n=1}^\infty$ admits a weak Cauchy subsequence.  By passing to a further subsequence if necessary, the sequence
\[\left(\frac{y_{2n+1}-y_{2n}}{\norm{y_{2n+1}-y_{2n}}_g}\right)_{n=1}^\infty\]
is normalized and weakly null.  Let us relabel if necessary so that in all cases $1\leq p<\infty$ we have found a normalized and weakly null basic sequence $(y_n)_{n=1}^\infty$ in $Y$.  By Bessaga-Pe\l czy\'{n}ski, we can pass to a subsequence if necessary so that $(y_n)_{n=1}^\infty$ is $(\sqrt{1+\epsilon})$-equivalent to a normalized block basic sequence $(y'_n)_{n=1}^\infty$ of the canonical basis of $g(w,p)$.  Now we apply Proposition \ref{block-basis-equivalent-lp} to find a block basis $(z'_k)_{k=1}^\infty$ of $(y'_n)_{n=1}^\infty$ which is $(\sqrt{1+\epsilon})$-equivalent to the canonical basis $(f_n)_{n=1}^\infty$ of $\ell_p$, where we write
$$z'_k=\sum_{n=p_k}^{p_{k+1}-1}a_ny'_n,\;\;\;k\in\mathbb{N}.$$
Define a corresponding basic sequence $(z_k)_{k=1}^\infty$ by the rule
$$z_k=\sum_{n=p_k}^{p_{k+1}-1}a_ny_n,\;\;\;k\in\mathbb{N}.$$
It is clear that $(z_k)_{k=1}^\infty$ is $(\sqrt{1+\epsilon})$-equivalent to $(z'_k)_{k=1}^\infty$, and hence $(1+\epsilon)$-equivalent to $(f_n)_{n=1}^\infty$.  It follows that $[z_k]_{k=1}^\infty$ is a subspace of $Y$ which is $(1+\epsilon)$-isomorphic to $\ell_p$.

(vi)  Let $Y$ be an infinite-dimensional closed subspace of $g(w,1)$.  By part (v), we can find a subspace $Z_1$ of $Y$ which is isomorphic to $\ell_1$.  By \cite[Theorem 1]{FW08}, every subspace isomorphic to $\ell_1$ of a Banach space $X$ with an unconditional basis contains a further subspace which is isomorphic to $\ell_1$ and complemented in $X$.  In particular, $Z_1$ contains a subspace $Z$ isomorphic to $\ell_1$ and complemented in $g(w,1)$.

(vii)  Let
$$y_n=\frac{\sum_{i=2^{n-1}}^{2^n-1}g_i}{\|\sum_{i=2^{n-1}}^{2^n-1}g_i\|_g},\;\;\;n\in\mathbb{N}.$$
By Proposition \ref{boundedly-complete}, $(y_n)_{n=1}^\infty$ is a block basic sequence satisfying the conditions of Theorem \ref{subsequence-equivalent-lp}.  Thus, we can find a subsequence $(y_{n_k})_{k=1}^\infty$ which is $(1+\epsilon)$-equivalent to the $\ell_p$ basis.  This subsequence is also 1-equivalent to the subsequence $(\hat{y}_{n_k})_{k=1}^\infty$ of the sequence $(\hat{y}_n)_{n=1}^\infty$ defined by Proposition \ref{1-dominated-blocks}. Again by Theorem \ref{subsequence-equivalent-lp}, the subsequence $(\hat{y}_{n_k})_{k=1}^\infty$ is $(1+\epsilon)$-complemented in $g(w,p)$.
\end{proof}







\section{$g(w,p)$ admits a unique subsymmetric basis}


In this section we show that for certain choices of $w\in\textbf{W}$, the canonical basis for $g(w,p)$, $1\leq p<\infty$, fails to be symmetric.  We also show that the canonical basis for $g(w,p)$ is always the unique (up to equivalence) subsymmetric basis for $g(w,p)$ (regardless of the choice of $w$).  Consequently, we exhibit examples of Banach spaces which admit a unique subsymmetric basis but not a symmetric one.

\begin{remark}It had previously been shown in \cite[\S6]{Sa04} that there exist Banach spaces, called Tirilman spaces, admitting a symmetric basis but failing to admit a subsymmetric one.  (In fact, those spaces do not even contain symmetric basic sequences!)  It is also easy to see based on existing results in \cite[\S3]{ACL73} (see Remark \ref{unique-subsymmetric-lorentz} below) that $d(w,p)$ admits a unique subsymmetric basis (which is also symmetric).  However, to the author's knowledge, the spaces $g(w,p)$ in Example \ref{not-symmetric} below are the first known examples of Banach spaces having both of these properties simultaneously---that is, admitting a unique subsymmetric basis but failing to admit a symmetric basis.\end{remark}


To prepare for our first task, we need a technical lemma.

\begin{lemma}\label{integral-bound}Let $0<\theta<1$ and $N\in\mathbb{N}$.  Then
\[\frac{(N+1)^{1-\theta}}{1-\theta}-\frac{1}{1-\theta}\leq\sum_{n=1}^Nn^{-\theta}\leq\frac{N^{1-\theta}}{1-\theta}-\frac{\theta}{1-\theta}.\]\end{lemma}

\begin{proof}Note that if $f:\mathbb{R}^+\to\mathbb{R}^+$ is nonincreasing and continuous on $[j,k]$, $j<k\in\mathbb{N}$, then
\[\sum_{n=j+1}^kf(n)\leq\int_j^kf(x)\;dx\leq\sum_{n=j}^{k-1}f(n).\]
Now apply this to $f(x)=x^{-\theta}$ over $[1,N]$.
%Thus,\[\sum_{n=1}^Nn^{-\theta}=1+\sum_{n=2}^Nn^{-\theta}\leq1+\int_1^Nx^{-\theta}\;dx=1+\left[\frac{x^{1-\theta}}{1-\theta}\right]_1^N=\frac{N^{1-\theta}}{1-\theta}-\frac{\theta}{1-\theta},\]and\[\sum_{n=1}^Nn^{-\theta}\geq\int_1^{N+1}x^{-\theta}\;dx=\left[\frac{x^{1-\theta}}{1-\theta}\right]_1^{N+1}=\frac{(N+1)^{1-\theta}}{1-\theta}-\frac{1}{1-\theta}.\]
\end{proof}



The proof of the following is essentially due to Garling (\cite{Ga68}).

\begin{example}\label{not-symmetric}Let $1\leq p<\infty$.  If $w=(n^{-\theta})_{n=1}^\infty\in\emph{\textbf{W}}$ for $\theta\in(0,1)$ then the canonical basis for $g(w,p)$ is subsymmetric but not symmetric.\end{example}

\begin{proof}For each $j\in\mathbb{N}$, let us set
\[z^{(j)}=(j^{-(1-\theta)/p},[j-1]^{-(1-\theta)/p},[j-2]^{-(1-\theta)/p},\cdots,1^{-(1-\theta)/p},0,0,\cdots)\]
and its nonincreasing rearrangement
\[y^{(j)}=(1^{-(1-\theta)/p},2^{-(1-\theta)/p},3^{-(1-\theta)/p},\cdots,j^{-(1-\theta)/p},0,0,0,\cdots).\]
Observe that
\[\norm{y^{(j)}}_{g}\geq\left(\sum_{n=1}^jn^{-1}\right)^{1/p}\to\infty\;\;\;\text{ as }\;\;\;n\to\infty.\]
Hence, $\sup_j\norm{y^{(j)}}_{g}=\infty$.

Next, we will show that $\sup_j\norm{z^{(j)}}_{g}<\infty$, and hence that the canonical basis is not symmetric.  To see this, first note that since the nonzero entries of $z^{(j)}$ are increasing, we must have $k\in\{1,\cdots,j\}$ such that
\[\norm{z^{(j)}}_{g}=\left(\sum_{n=1}^k(k+1-n)^{-(1-\theta)}n^{-\theta}\right)^{1/p}.\]
We claim that
\begin{equation}\label{z-estimate}\sup_{k\in\mathbb{N}}=\sum_{n=1}^k(k+1-n)^{-(1-\theta)}n^{-\theta}<\infty\end{equation}
whence it shall follow that $\sup_j\norm{z^{(j)}}_{g}<\infty$ as desired.

Consider the case where $k\geq 3$ and is odd, and let us give the estimate
\begin{multline*}
\sum_{n=1}^k(k+1-n)^{-(1-\theta)}n^{-\theta}\\=\sum_{n=1}^{(k+1)/2}(k+1-n)^{-(1-\theta)}n^{-\theta}+\sum_{n=(k+3)/2}^k(k+1-n)^{-(1-\theta)}n^{-\theta}
\\\leq\left(\frac{k+1}{2}\right)^{-(1-\theta)}\sum_{n=1}^{(k+1)/2}n^{-\theta}+\left(\frac{k+3}{2}\right)^{-\theta}\sum_{n=(k+3)/2}^k(k+1-n)^{-(1-\theta)}
\\=\left(\frac{k+1}{2}\right)^{-(1-\theta)}\sum_{n=1}^{(k+1)/2}n^{-\theta}+\left(\frac{k+3}{2}\right)^{-\theta}\sum_{n=1}^{(k-1)/2}n^{-(1-\theta)}\\\leq 2\left(\frac{k+1}{2}\right)^{-(1-\theta)}\sum_{n=1}^{(k+1)/2}n^{-\theta}.
\end{multline*}
Observe that by Lemma \ref{integral-bound} we have
\[\sum_{n=1}^{(k+1)/2}n^{-\theta}\leq\frac{1}{1-\theta}\left(\frac{k+1}{2}\right)^{1-\theta}
\]
so that
\[2\left(\frac{k+1}{2}\right)^{-(1-\theta)}\sum_{n=1}^{(k+1)/2}n^{-\theta}\leq\frac{2}{1-\theta}.\]
The even case is handled similarly, and this proves \eqref{z-estimate}.\end{proof}


\begin{remark}It was shown in \cite[Theorem 6]{ACL73} that $d(w,p)$ admits exactly two nonequivalent symmetric basic sequences if and only if the weights $w=(w_n)_{n=1}^\infty\in\emph{\textbf{W}}$ satisfy the condition
\begin{equation}\label{exactly-two}\sup_{j,k\in\mathbb{N}}\frac{\sum_{n=1}^{jk}w_n}{\left(\sum_{n=1}^jw_n\right)\left(\sum_{n=1}^kw_n\right)}<\infty.\end{equation}
Observe that, by Lemma \ref{integral-bound}, the weight sequence $w=(n^{-\theta})_{n=1}^\infty\in\emph{\textbf{W}}$, $\theta\in(0,1)$, given in Example \ref{not-symmetric} satisfies \eqref{exactly-two}.\end{remark}

%\begin{proof}Let us begin with the observation that, by l'H\^{o}pital,\[\lim_{j\to\infty}\frac{j^{1-\theta}}{(j+1)^{1-\theta}-1}=\lim_{j\to\infty}\frac{(1-\theta)j^{-\theta}}{(1-\theta)(j+1)^{-\theta}}=\lim_{j\to\infty}\left(\frac{j+1}{j}\right)^\theta=1.\]So we can find $C\geq 1$ such that\[\frac{j^{1-\theta}}{(j+1)^{1-\theta}-1}\leq C\]and hence\[j^{1-\theta}C^{-1}\leq(j+1)^{1-\theta}-1\]for all $j\in\mathbb{N}$.  Together with Lemma \ref{integral-bound}, we now have the estimate\begin{multline*}\frac{\sum_{n=1}^{jk}w_n}{\left(\sum_{n=1}^jw_n\right)\left(\sum_{n=1}^kw_n\right)}\leq\frac{\displaystyle\frac{(jk)^{1-\theta}-\theta}{1-\theta}}{\displaystyle\frac{[(j+1)^{1-\theta}-1][(k+1)^{1-\theta}-1]}{(1-\theta)^2}}\\=\frac{[(jk)^{1-\theta}-\theta][1-\theta]}{[(j+1)^{1-\theta}-1][(k+1)^{1-\theta}-1]}\leq\frac{(jk)^{1-\theta}}{[j^{1-\theta}C^{-1}][k^{1-\theta}C^{-1}]}=C^2\end{multline*}for all $j,k\in\mathbb{N}$.\end{proof}

Before continuing, we recall the following result from \cite{ACL73}, which shall be needed momentarily.

\begin{proposition}[{\cite[Corollary 2]{ACL73}}]\label{Corollary-2}Let $1\leq p<\infty$ and $w=(w_n)_{n=1}^\infty\in\emph{\textbf{W}}$, and let $(d_n)_{n=1}^\infty$ denote the canonical basis of $d(w,p)$.  Every seminormalized basic sequence in $d(w,p)$ admits a subsequence dominating $(d_n)_{n=1}^\infty$.\end{proposition}

Our next task requires the following analogous result for $g(w,p)$.

\begin{proposition}\label{symmetric-dominates-Lorentz}Let $1\leq p<\infty$ and $w=(w_n)_{n=1}^\infty\in\emph{\textbf{W}}$, let $(d_n)_{n=1}^\infty$ denote the canonical basis of $d(w,p)$, and let $(g_n)_{n=1}^\infty$ denote the canonical basis of $g(w,p)$.  Then every symmetric normalized block sequence of $(g_n)_{n=1}^\infty$ dominates $(d_n)_{n=1}^\infty$.\end{proposition}

\begin{proof}Fix any symmetric normalized block sequence $(y_n)_{n=1}^\infty$ of $(g_n)_{n=1}^\infty$ in $g(w,p)$.  For any finite sequence of scalars $(b_n)_{n=1}^N$, $N\in\mathbb{N}$, we have
$$\norm{\sum_{n=1}^N b_nd_n}_{d}
=\left(\sum_{n=1}^N |b_{\sigma(n)}|^pw_n\right)^{1/p}$$
for some permutation $\sigma$ of $\{1,\cdots,N\}$.  However, due to symmetry together with Corollary \ref{dominates-or-equivalent}, we can find $C\in[1,\infty)$ such that $(y_n)_{n=1}^\infty$ is $C$-equivalent to all its subsequences, and $C$-dominates $(g_n)_{n=1}^\infty$.  Thus,
\begin{multline*}
\norm{\sum_{n=1}^N b_nd_n}_{d}
=\left(\sum_{n=1}^N |b_{\sigma(n)}|^pw_n\right)^{1/p}
\\\leq\|\sum_{n=1}^Nb_{\sigma(n)}g_n\|_g
\leq C\|\sum_{n=1}^Nb_{\sigma(n)}y_n\|_g
\leq C^2\|\sum_{n=1}^Nb_ny_n\|_g
\end{multline*}
%\[\norm{\sum_{n=1}^N b_{\sigma(n)}y_n}_{g}^p\geq\frac{1}{C^{2p}}\norm{\sum_{n=1}^N b_{\sigma(n)}g_n}_{g}^p\geq\frac{1}{C^{2p}}\sum_{n=1}^N|b_{\sigma(n)}|^pw_n.\]
It follows that $(y_n)_{n=1}^\infty$ $C^2$-dominates $(d_n)_{n=1}^\infty$.\end{proof}

The next theorem is based on \cite[Theorem 4]{ACL73}, wherein an analogous result for subspaces with symmetric bases was proved for $d(w,p)$.  However, our proof here is slightly different.

\begin{theorem}\label{unique-subsymmetric}Let $1\leq p<\infty$ and $w=(w_n)_{n=1}^\infty\in\emph{\textbf{W}}$, and let $(g_n)_{n=1}^\infty$ denote the canonical basis of $g(w,p)$.  If $(y_n)_{n=1}^\infty$ is a subsymmetric basic sequence in $g(w,p)$ and $Y=[y_n]_{n=1}^\infty$ then every subsymmetric basis for $Y$ is equivalent to $(y_n)_{n=1}^\infty$.  In particular, every subsymmetric basis for $g(w,p)$ is equivalent to $(g_n)_{n=1}^\infty$.\end{theorem}

\begin{proof}Assume $Y$ is not isomorphic to $\ell_p$, else we are done.  Now let $(u_n)_{n=1}^\infty$ be any subsymmetric basis for $Y$.  By Proposition \ref{subsymmetric-weakly-null} together with the fact that $g(w,p)$ and hence $Y$ contains no copy of $\ell_1$, the sequence $(u_n)_{n=1}^\infty$ must be weakly null, and the same goes for $(y_n)_{n=1}^\infty$.  Thus, we may apply Bessaga-Pe\l czy\'{n}ski and the Principle of Small Perturbations to obtain normalized blocks
\[\hat{u}_i=\sum_{n=p_i}^{p_{i+1}-1}a_ny_n,\;\;\;\hat{y}_i=\sum_{n=q_i}^{q_{i+1}-1}b_ng_n,\;\;\;z_i=\sum_{n=r_i}^{r_{i+1}-1}c_nu_n,\;\;\;i\in\mathbb{N},\]
such that $(\hat{u}_n)_{n=1}^\infty\approx(u_n)_{n=1}^\infty$ and $(\hat{y}_n)_{n=1}^\infty\approx(y_n)_{n=1}^\infty\approx(z_i)_{i=1}^\infty$.  For convenience, we may simply assume $\hat{y}_n=y_n$ and $\hat{u}_n=u_n$ for $n\in\mathbb{N}$.

Let us now write
\[z_k=\sum_{i=r_k}^{r_{k+1}-1}\sum_{j=p_i}^{p_{i+1}-1}\sum_{n=q_j}^{q_{j+1}-1}c_ia_jb_ng_n.\]
Notice that by Theorem \ref{subsequence-equivalent-lp} we cannot have $\lim_{j\to\infty}|a_j|=0$, $\lim_{n\to\infty}|b_n|=0$, or $\lim_{i\to\infty}|c_i|=0$, since otherwise, together with subsymmetry, $(z_k)_{k=1}^\infty$ would be equivalent to the canonical basis of $\ell_p$.  Together with subsymmetry it follows that, passing to subsequences if necessary, we may assume the coefficients $(a_j)_{j=1}^\infty$, $(b_n)_{n=1}^\infty$, and $(c_i)_{i=1}^\infty$ each satisfy property \eqref{4}.  By Proposition \ref{subsymmetric-dominates}, this means
$$
(y_n)_{n=1}^\infty\lesssim(u_n)_{n=1}^\infty\lesssim(z_n)_{n=1}^\infty\approx(y_n)_{n=1}^\infty.
$$\end{proof}

\begin{remark}\label{unique-subsymmetric-lorentz}It is easy to see based on the analogous results in \cite{ACL73} that Theorem \ref{unique-subsymmetric} also holds for $d(w,p)$ in place of $g(w,p)$.\end{remark}

\begin{corollary}\label{unique-subsymmetric-corollary}Each of the spaces constructed in Example \ref{not-symmetric} admits a unique (up to equivalence) subsymmetric basis, but not a symmetric one.\end{corollary}

%\begin{theorem}\label{unique-subsymmetric}Let $1\leq p<\infty$ and $w=(w_n)_{n=1}^\infty\in\emph{\textbf{W}}$, and let $(g_n)_{n=1}^\infty$ denote the canonical basis of $g(w,p)$.  Then every subsymmetric basis for $g(w,p)$ is equivalent to $(g_n)_{n=1}^\infty$.\end{theorem}
%
%\begin{proof}Suppose $(u_n)_{n=1}^\infty$ is a subsymmetric basis for $g(w,p)$.  By \cite[Proposition 1]{ACL73}, every symmetric basic sequence in a Banach space is either weakly null or else equivalent to the canonical basis of $\ell_1$.  Since $g(w,p)$ is  not isomorphic to $\ell_p$, then $(u_n)_{n=1}^\infty$ must be weakly null.  We may now apply Bessaga-Pe\l czy\'{n}ski to obtain blocks
%\[y_i=\sum_{n=p_i}^{p_{i+1}-1}a_ng_n,\;\;\;i\in\mathbb{N}\]
%such that $(y_i)_{i=1}^\infty$ is equivalent to $(u_n)_{n=1}^\infty$.  By the Principle of Small Perturbations together with subsymmetry of $(y_i)_{i=1}^\infty$ we may assume that $(y_i)_{i=1}^\infty$ is normalized.  Since $(y_i)_{i=1}^\infty$ is weakly null, Bessaga-Pe\l czy\'{n}ski again delivers us blocks
%\[z_k=\sum_{i=q_k}^{q_{k+1}-1}b_iy_i,\;\;\;k\in\mathbb{N}\]
%such that $(z_k)_{k=1}^\infty$ is equivalent to $(g_n)_{n=1}^\infty$.  As before we may assume that $(z_k)_{k=1}^\infty$ is also normalized.  Let us now write
%\[z_k=\sum_{i=q_k}^{q_{k+1}-1}\sum_{n=p_i}^{p_{i+1}-1}b_ia_ng_n=\sum_{n=p_{q_k}}^{p_{q_{k+1}}-1}c_ng_n,\]
%where $c_n=b_ia_n$ for all $p_i\leq n\leq p_{i+1}-1$ and $i\in\mathbb{N}$.  Notice that by Theorem \ref{subsequence-equivalent-lp} we cannot have $\lim_{n\to\infty}|c_n|=0$, since otherwise, together with subsymmetry, $(z_k)_{k=1}^\infty$ would be equivalent to the canonical basis of $\ell_p$.  By unconditionality of $(g_n)_{n=1}^\infty$ and normalization of $(y_i)_{i=1}^\infty$, we have $|a_n|\leq 1$, and also $n\to\infty$ as $i\to\infty$, where $p_i\leq n\leq p_{i+1}-1$.  Hence, we cannot have $\lim_{i\to\infty}|b_i|=0$ either, since in that case we would have $|c_n|=|b_ia_n|\leq|b_i|\to 0$.  It follows that the coefficients $(b_i)_{i=1}^\infty$ satisfy property \eqref{4}, so that by Proposition \ref{subsymmetric-dominates}, $(z_k)_{k=1}^\infty\geq(y_i)_{i=1}^\infty$.  A similar argument shows that $(g_n)_{n=1}^\infty\leq(y_i)_{i=1}^\infty$.  We now obtain
%\[(u_n)_{n=1}^\infty\approx(y_i)_{i=1}^\infty\leq(z_k)_{k=1}^\infty\approx(g_n)_{n=1}^\infty\leq(y_i)_{i=1}^\infty\approx(u_n)_{n=1}^\infty\]
%so that $(u_n)_{n=1}^\infty\approx(g_n)_{n=1}^\infty$.\end{proof}
%
%\begin{remark}Similar to the proof of Theorem \ref{unique-subsymmetric}, we can show that $d(w,p)$ always fails to admit a basis which is subsymmetric but not symmetric.  Indeed, if $(y_n)_{n=1}^\infty$ is a subsymmetric basis then we could find a normalized block basic sequence $(y_i)_{n=1}^\infty$ of $(d_n)_{n=1}^\infty$ which is equivalent to $(u_n)_{n=1}^\infty$, and then a normalized block basic squence $(z_k)_{k=1}^\infty$ of $(y_i)_{i=1}^\infty$ which is equivalent to $(d_n)_{n=1}^\infty$.  As all block coefficients must satisfy property \eqref{4}, we get
%\[(d_n)_{n=1}^\infty\approx(z_k)_{k=1}^\infty\geq(y_i)_{i=1}^\infty\geq(d_n)_{n=1}^\infty\]
%so that $(d_n)_{n=1}^\infty\approx(y_i)_{i=1}^\infty\approx(u_n)_{n=1}^\infty$.  In particular, while it was already known from \cite[Theorem 4]{ACL73} that every symmetric basis for $d(w,p)$ is equivalent, so too is every subsymmetric basis.\end{remark}


%Despite having found in Example \ref{not-symmetric} certain choices of $w\in\textbf{W}$ for which the canonical basis $(g_n)_{n=1}^\infty$ of $g(w,p)$ is not symmetric, we do not whether there might be other choices of $w$ for which $(g_n)_{n=1}^\infty$ is symmetric.  To close this section, we give some equivalent conditions where this is the case.

We close this section by exploring the following question.

\begin{question}\label{question-symmetric}Do there exist $w\in\emph{\textbf{W}}$ and $1\leq p<\infty$ so that $g(w,p)$ admits a symmetric basis?\end{question}

\noindent It follows from Corollary \ref{unique-subsymmetric-corollary} that not every $w\in\textbf{W}$ yields a positive answer to Question \ref{question-symmetric}, but we do not know whether some other $w\in\textbf{W}$ might do the trick.

Theorem \ref{unique-subsymmetric} tells us that $g(w,p)$ admits a symmetric basis if and only if its canonical basis is symmetric.  The next result lays out some other equivalent conditions.

\begin{theorem}\label{tfae}Let $1\leq p<\infty$ and $w=(w_n)_{n=1}^\infty\in\emph{\textbf{W}}$, let $(d_n)_{n=1}^\infty$ denote the canonical basis of $d(w,p)$, and let $(g_n)_{n=1}^\infty$ denote the canonical basis of $g(w,p)$.  Then the following are equivalent.
\begin{itemize}\item[(i)]  $g(w,p)$ admits a symmetric basis.
\item[(ii)]  $(g_n)_{n=1}^\infty$ is equivalent to $(d_n)_{n=1}^\infty$ (and in particular is symmetric).
\item[(iii)]  $g(w,p)$ and $d(w,p)$ are isomorphic.
\item[(iv)]  $d(w,p)$ contains a subspace isomorphic to $g(w,p)$.
\item[(v)]  The formal inclusion map
\[I_{d,g}\in\mathcal{L}(d(w,p),g(w,p)),\;\;\;I_{d,g}d_n=g_n\;\forall n\in\mathbb{N},\]
preserves a copy of $d(w,p)$ which is complemented in $g(w,p)$.  More precisely, there exists a subspace $Y$ of $d(w,p)$ for which $Y\approx d(w,p)\approx I_{d,g}Y$, and such that $I_{d,g}Y$ is complemented in $g(w,p)$ and $I_{d,g}|_Y$ is bounded below.\end{itemize}\end{theorem}

\begin{proof}Let us begin by noting that (ii)$\Rightarrow$(i), (ii)$\Rightarrow$(iii), (iii)$\Rightarrow$(iv), and (ii)$\Rightarrow$(v) are all trivial.

(i)$\Rightarrow$(ii):  Note that $(g_n)_{n=1}^\infty\lesssim(d_n)_{n=1}^\infty$ by a direct application of their definitions.  By Theorem \ref{unique-subsymmetric}, $(g_n)_{n=1}^\infty$ is symmetric, and so by Proposition \ref{symmetric-dominates-Lorentz} we have $(d_n)_{n=1}^\infty\lesssim(g_n)_{n=1}^\infty$.  It follows that $(d_n)_{n=1}^\infty\approx(g_n)_{n=1}^\infty$.

(iv)$\Rightarrow$(ii):  Since $(g_n)_{n=1}^\infty$ is weakly null by Theorem \ref{main}, we obtain via Bessaga-Pe\l czy\'{n}ski a seminormalized $(d_n)_{n=1}^\infty$-block basic sequence $(y_n)_{n=1}^\infty$ in $d(w,p)$ which is equivalent to $(g_n)_{n=1}^\infty$, and hence dominated by $(d_n)_{n=1}^\infty$.  Since $(y_n)_{n=1}^\infty$ is subsymmetric, by Proposition \ref{Corollary-2} we get $(y_n)_{n=1}^\infty\gtrsim(d_n)_{n=1}^\infty$.  Combining these gives
$$
(y_n)_{n=1}^\infty\approx(g_n)_{n=1}\infty\lesssim(d_n)_{n=1}^\infty\lesssim(y_n)_{n=1}^\infty.
$$
%that $(g_n)_{n=1}^\infty$ and $(d_n)_{n=1}^\infty$ are equivalent.

(v)$\Rightarrow$(ii):  Since $I_{d,g}Y$ is complemented in $g(w,p)$ and isomorphic to $d(w,p)$, we could extend an isomorphism $\widehat{B}:I_{d,g}Y\to d(w,p)$ to an operator (not necessarily an isomorphism anymore) $B:g(w,p)\to d(w,p)$.  By \cite[Theorem 5.5]{KPSTT12}, an operator in $\mathcal{L}(d(w,p))$ fails to preserve a copy of $d(w,p)$ if and only if it sends the canonical basis to a norm-null sequence.  So, $BI_{d,g}d_n\not\to 0$, which means we could find some subsequence $(Bg_{n_k})_{k=1}^\infty$ which is seminormalized.  As $(g_n)_{n=1}^\infty$ is weakly null by Theorem \ref{main}, so is $(Bg_{n_k})_{k=1}^\infty$.  Pass to a further subsequence if necessary so that, by applying Bessaga-Pe\l czy\'{n}ski, $(Bg_{n_k})_{k=1}^\infty$ is seminormalized basic, and equivalent to a block basis of $(d_n)_{n=1}^\infty$.  By Proposition \ref{Corollary-2} we may assume $(Bg_{n_k})_{k=1}^\infty\gtrsim(d_n)_{n=1}^\infty$.  Hence,
\[(d_n)_{n=1}^\infty\lesssim(Bg_{n_k})_{k=1}^\infty\lesssim(g_{n_k})_{k=1}^\infty\approx(g_n)_{n=1}^\infty\lesssim(d_n)_{n=1}^\infty.\]\end{proof}







\begin{remark}In the proof of (v)$\Rightarrow$(ii), we actually showed that if $(g_n)_{n=1}^\infty\not\approx(d_n)_{n=1}^\infty$ then for no $B\in\mathcal{L}(g(w,p),d(w,p))$ does the composition operator $BI_{d,g}$ preserve a copy of $d(w,p)$, regardless of whether or not it passes that copy through a complemented subspace of $g(w,p)$.  On the other hand, it is clear by considering the proof to Proposition \ref{block-basis-equivalent-lp} that the identity operator on $\ell_p$ factors through $I_{d,g}$, so that $I_{d,g}$ is never strictly singular.\end{remark}



%
%As a corollary to Example \ref{not-symmetric} and Theorems \ref{main} and \ref{tfae}, we obtain, for each $1\leq p<\infty$, a family of complementably $\ell_p$-saturated Banach spaces which admit a unique (up to equivalence) subsymmetric basis but do not admit a symmetric basis





\section{Uniform convexity and superreflexivity}

We have seen that, in some cases at least, $g(w,p)$ is not isomorphic to $d(w,p)$.  We do not know whether this is true in all cases.  However, we could say some things about isometries.  Indeed, it is clear that $g(w,p)$ is never isometric in the natural way to $d(w,p)$.  Just consider the minimum $N\in\mathbb{N}$ such that $w_{N+1}<1$, so that, letting
\[0<\alpha<\left(\frac{1-w_{N+1}}{N}\right)^{1/p}\;\;\;\text{ and }\;\;\;u=(\underbrace{\alpha,\cdots,\alpha}_{N\text{ times}},1,0,0,0,\cdots),\]
we have
\[\norm{u}_{g}=1<\norm{u}_d.\]
Using a roundabout method, we can even show that, for certain choices of $w\in\textbf{W}$, more general than in Example \ref{not-symmetric}, $g(w,p)$ does not isometrically embed into $d(w,p)$ for any $1<p<\infty$ (although, we do not know whether it might isomorphically embed).

The method we will use is to first prove that $g(w,p)$ is never uniformly convex, and thus cannot embed isometrically into $d(w,p)$ whenever the latter space is.  Recall that a Banach space $X$ is {\bf uniformly convex} just in case for each $\epsilon>0$ there is $\delta=\delta(\epsilon)>0$ such that if $u,v\in X$ with $\|u\|_X=\|v\|_X=1$ and $\norm{u-v}_X\geq\epsilon$ then $\frac{1}{2}\norm{u+v}_X\leq 1-\delta$.

\begin{proposition}The space $g(w,p)$ fails to be uniformly convex for any $1\leq p<\infty$ and any $w=(w_n)_{n=1}^\infty\in\textbf{\emph{W}}$.\end{proposition}

\begin{proof}Let us define, for $j\in\mathbb{N}$,
\[\alpha_j=\left(\frac{1-w_{j+1}}{\sum_{n=1}^jw_n}\right)^{1/p}.\]
Now let us consider, for each $j\in\mathbb{N}$, the vectors
\[u^{(j)}=(\underbrace{\alpha_j,\cdots,\alpha_j}_{j\text{ times}},1,0,0,0,\cdots)\in g(w,p)\]
and
\[v^{(j)}=(\underbrace{\alpha_j,\cdots,\alpha_j}_{j\text{ times}},0,1,0,0,0,\cdots)\in g(w,p).\]
Observe that
\begin{multline*}\frac{1}{2}\norm{u^{(j)}+v^{(j)}}_{g}
\geq\frac{1}{2}\left[\sum_{n=1}^{j}(2\alpha_{j})^pw_n+w_{j+1}+w_{j+2}\right]^{1/p}
\\=\left[\alpha_j^p\sum_{n=1}^jw_n+\frac{1}{2^p}(w_{j+1}+w_{j+2})\right]^{1/p}\\=\left[1-w_{j+1}+\frac{1}{2^p}(w_{j+1}+w_{j+2})\right]^{1/p}\to 1\end{multline*}
as $j\to\infty$.  To show that $g(w,p)$ fails to be uniformly convex, then, it suffices to find a subsequence $(u^{(j_k)})_{k=1}^\infty$ such that $\norm{u^{(j_k)}}_{g}=\norm{v^{(j_k)}}_{g}=1$ and $\norm{u^{(j_k)}-v^{(j_k)}}_{g}>1$ for all $k\in\mathbb{N}$.

Due to $(w_j)_{j=1}^\infty\in c_0\setminus\ell_1$ we have
\[\lim_{j\to\infty}\alpha_j=0.\]
Hence, we could find a subsequence $(\alpha_{j_k})_{k=1}^\infty$ such that
\[\alpha_{j_k}\leq\min_{i\leq j_k}\alpha_i,\;\;\;k\in\mathbb{N}.\]
%Indeed, set $j_1=1$, and if $j_k$ has been defined for some $k\in\mathbb{N}$ then let $j_{k+1}>j_k$ be minimal such that $\alpha_{j_{k+1}}\leq\min_{i\leq j_k}\alpha_i$.  These indices will give us the subsequences $(u^{(j_k)})_{k=1}^\infty$ and $(v^{(j_k)})_{k=1}^\infty$ we desire.

Now, fix any $k\in\mathbb{N}$.  By definition of $g(w,p)$ and due to $w_1=1$, either $\norm{u^{(j_k)}}_{g}=1$, or else we could find $i\in\{1,\cdots,j_k\}$ with
\[1\leq\norm{u^{(j_k)}}_{g}^p=\sum_{n=1}^iw_n\alpha_{j_k}^p+w_{i+1}\leq\sum_{n=1}^iw_n\alpha_i^p+w_{i+1}=1\]
so that $\norm{u^{(j_k)}}_{g}=\norm{v^{(j_k)}}_{g}=1$ anyway.  Observing that
\[\norm{u^{(j_k)}-v^{(j_k)}}_{g}=(w_1+w_2)^{1/p}>1\]
finishes the proof.\end{proof}

Note that uniform convexity is an isometric property, and that one uniformly convex space might be isomorphic (but not isometric) to a space which is not uniformly convex.  In light of this, let us say that a Banach space $X$ is {\bf uniformly convexifiable} whenever there exists an equivalent norm on $X$ for which it is uniformly convex, or, equivalently, whenever it is isomorphic to a uniformly convex space.  Uniform convexifiability has been famously identified with superreflexivity, as follows.  Recall that a Banach space $Y$ is {\bf finitely representable} in a Banach space $X$ whenever for each $\epsilon>0$ and each finite-dimensional subspace $F$ of $Y$, there exists a finite-dimensional subspace $E$ of $X$ such that $E$ and $F$ are $(1+\epsilon)$-isomorphic, i.e. such that there exists an isomorphism $U:E\to F$ satisfying $\norm{U}\norm{U^{-1}}\leq 1+\epsilon$.  Then, a Banach space $X$ is called {\bf superreflexive} just in case every Banach space finitely representable in $X$ is reflexive.  It was shown in \cite{En73} that a Banach space is superreflexive if and only if it is uniformly convexifiable.

Having shown that $g(w,p)$ is never uniformly convex, and in light of the above identification between superreflexivity and uniform convexifiability, the next natural question to ask is, given $1<p<\infty$, can we ever choose $w\in\textbf{W}$ so that $g(w,p)$ is superreflexive?  This looks more difficult to answer, but due to the work in \cite{Al75}, we have a bit of a head start by ruling out a few cases.

\begin{theorem}[{\cite[Theorem 1]{Al75}}]\label{Lorentz-uniformly-convex}Let $1<p<\infty$ and $w=(w_n)_{n=1}^\infty\in\emph{\textbf{W}}$.    Then the following are equivalent.
\begin{itemize}\item[(I)] $d(w,p)$ is uniformly convex.
\item[(II)] $d(w,p)$ is uniformly convexifiable (equivalently, superreflexive).
\item[(III)] \[\inf_{n\in\mathbb{N}}\frac{\sum_{i=1}^{2n}w_i}{\sum_{i=1}^nw_i}>1.\]
\item[(IV)] There exist constants $A>0$ and $B>0$ such that
\[Bw_n\leq\frac{1}{n}\sum_{i=1}^nw_i\leq Aw_n\]
for all $n\in\mathbb{N}$.\end{itemize}\end{theorem}

\begin{remark}The choices $w=(n^{-\theta})_{n=1}^\infty\in\emph{\textbf{W}}$, $0<\theta<1$, from Example \ref{not-symmetric} satisfy the equivalent conditions (III) and (IV) above.\end{remark}

\noindent As clearly every Banach space isometric to a subspace of uniformly convex is itself uniformly convex, we have the following immediate corollary.

\begin{corollary}Let $1<p<\infty$ and $w=(w_n)_{n=1}^\infty\in\emph{\textbf{W}}$.  If any of the equivalent conditions (I)-(IV) in Theorem \ref{Lorentz-uniformly-convex} hold, then no subspace of $d(w,p)$ is isometric to $g(w,p)$.\end{corollary}

Let us return to the question of superreflexivity.  The following lemma, which we will need shortly, was originally stated with only symmetric bases in mind, but a quick look at the proof shows that subsymmetry is quite sufficient.

\begin{lemma}[{\cite[Lemma 4]{Al75}}]\label{Lemma-4}Let $(x_n)_{n=1}^\infty$ be a subsymmetric basis for a Banach space $X$.  Suppose there are increasing sequences $(n_i)_{i=1}^\infty\subseteq\mathbb{N}$ and $(k_i)_{i=1}^\infty\subseteq\mathbb{N}$ and a uniform constant $M>0$ such that
\[\sup_{i\in\mathbb{N}}\frac{\norm{\sum_{n=1}^{n_ik_i}x_n}_X}{\norm{\sum_{n=1}^{n_i}x_n}_X}\leq M.\]
Then $X$ is not uniformly convexifiable (equivalently, not superreflexive).\end{lemma}

\noindent The author proved it by finding $k_i$-dimensional subspaces of $X$, defined as the span of vectors
\[v_m=\frac{1}{\norm{\sum_{i=1}^{n_i}x_i}}\sum_{j=(m-1)n_i+1}^{mn_i}x_i,\;\;\;m=1.\cdots,k_i,\]
which are uniformly isomorphic to $\ell_\infty^{k_i}$.  Indeed, assuming 1-subsymmetry for convenience (so that the $v_m$'s are normalized instead of just seminormalized), any sequence $(\alpha_m)_{m=1}^{k_i}$ of scalars gives us
\begin{multline*}\sup_{1\leq m\leq k_i}|\alpha_m|\leq\norm{\sum_{m=1}^{k_i}\alpha_mv_m}_X\leq\sup_{1\leq m\leq k_i}|\alpha_m|\cdot\norm{\sum_{m=1}^{k_i}v_m}_X
\\\leq\sup_{1\leq m\leq k_i}|\alpha_m|\cdot\sup_{j\in\mathbb{N}}\frac{\norm{\sum_{n=1}^{n_jk_j}x_n}_X}{\norm{\sum_{n=1}^{n_j}x_n}_X}\leq M\sup_{1\leq m\leq k_i}|\alpha_m|.\end{multline*}

We also need one more lemma, from the same paper.

\begin{lemma}[{\cite[Lemma 5]{Al75}}]\label{Lemma-5}Let $w=(w_n)_{n=1}^\infty\in\emph{\textbf{W}}$.  If
\[\inf_{n\in\mathbb{N}}\frac{\sum_{i=1}^{2n}w_i}{\sum_{i=1}^nw_i}=1\]
then for every $\epsilon>0$ there exist strictly increasing sequences $(n_i)_{i=1}^\infty\subseteq\mathbb{N}$ and $(k_i)_{i=1}^\infty\subseteq\mathbb{N}$ such that
\[\sup_{i\in\mathbb{N}}\frac{\sum_{n=1}^{n_ik_i}w_n}{\sum_{m=1}^{n_i}w_n}\leq 1+\epsilon.\]
\end{lemma}

\begin{corollary}Let $1<p<\infty$ and $w=(w_n)_{n=1}^\infty\in\emph{\textbf{W}}$.  If any of the equivalent conditions (I)-(IV) in Theorem \ref{Lorentz-uniformly-convex} fail to hold then $g(w,p)$ is not uniformly convexifiable.  In particular, in this case, $g(w,p)$ is reflexive but not superreflexive.\end{corollary}

\begin{proof}Note that if condition (III) fails to hold then we can apply Lemma \ref{Lemma-5}, and hence also Lemma \ref{Lemma-4}, to conclude that $g(w,p)$ is not uniformly convexifiable.  The last statement is just due to uniform convexifiability being equivalent to superreflexivity (cf. \cite{En73}), together with Theorem \ref{main}.\end{proof}


\begin{thebibliography}{999}


\bibitem[AK06]{AK06}
Fernando Albiac and Nigel J. Kalton.
\newblock {\it Topics in Banach Space Theory.}
\newblock Graduate Texts in Mathematics,
\newblock Springer Inc. (2006), ISBN 978-0387-28141-4.


\bibitem[Al75]{Al75}
Zvi Altshuler.
``Uniform convexity in Lorentz sequence spaces,''
{\it Israel Journal of Mathematics}, Volume 20, Nos. 3-4 (1975), pp260-274.

\bibitem[ACL73]{ACL73}
Zvi Altshuler, Peter G. Casazza, Bor-Luh Lin.
``On symmetric basic sequences in Lorentz sequence spaces,''
{\it Israel Journal of Mathematics}, Volume 15, Issue 2 (June, 1973), pp140-55.

%
%
%
%\bibitem[BCFW15]{BCFW15}
%Kevin Beanland, Ryan Causey, Daniel Freeman, Ben Wallis.
%\newblock ``Classes of operators determined by ordinal indices''
%\newblock (preprint, 2015).  \verb|arXiv:1507.06285v2 [math.FA]|
%
%
%    
%
%\bibitem[BF03]{BF03}
%\newblock John J. Benedetto and Matthew Fickus.
%\newblock ``Finite Normalized Tight Frames.''
%\newblock \emph{Advances in Computational Mathematics} 18: 357–385, 2003.
%
%\bibitem[CK12]{CK12}
%Peter G. Casazza and Gitta Kutyniok.
%\newblock {\it Finite Frames:  Theorey and Applications}
%\newblock (2012).
%
%

\bibitem[CL74]{CL74}
P.G. Casazza, Bor-Luh Lin.
``On symmetric basic sequences in lorentz sequence spaces II,''
{\it Israel Journal of Mathematics} Volume 17, Issue 2 (December 1974), pp191-218.


\bibitem[DOSZ11]{DOSZ11}
Stephen J. Dilworth, E. Odell, Thomas Schlumprecht, Andras Zs\'ak.
``Renormings and Symmetry Properties of 1-greedy bases,''
{\it J. Approx Theory} 163 (2011), pp1049-1075.

%\bibitem[Fab01]{Fab01}
%Mari\'an J. Fabian, Petr Habala, Petr H\'ajek, Vicente Montesinos Santaluc\'ia, Jan Pelant, and V\'aclav Zizler.
%\newblock {\it Functional Analysis and Infinite-Dimensional Geometry}
%\newblock (2001).
%

\bibitem[En73]{En73}
Per Enflo.
``Banach spaces which can be given an equivalent uniformly convex norm,''
{\it Israel J. Math.} 13 (1973), pp281–288.


\bibitem[FW08]{FW08}
Carlos Finol and Marek Wójtowicz.
``Complemented copies of $\ell_1$ in Banach spaces with an unconditional basis,''
{\it Journal of Mathematical Analysis and Applications} 342:1 (1 June 2008), pp83-88.


%
%\bibitem[Fe80]{Fe80}
%Moshe Feder.
%``On subspaces of spaces with an unconditional basis and spaces of operators,''
%{\it Illinois Journal of Mathematics} Volume 24, Number 2 (summer, 1980), pp196-205.
%

%\bibitem{Casazza}
%\newblock Peter G. Casazza, Gitta Kutyniok and Friedrich Philipp.
%\newblock \emph{Introduction to Finite Frame Theory.}
%\newblock (internet document)
%\newblock \begin{verbatim}http://www.springer.com/cda/content/document\end{verbatim}\begin{verbatim}/cda_downloaddocument/9780817683726-c1.pdf\end{verbatim}
%

\bibitem[Ga68]{Ga68}
D.J.H. Garling.
``Symmetric bases of locally convex spaces,''
{\it Studia Mathematica}, Vol. 30, No. 2 (1968), pp163-81.

\bibitem[KPSTT12]{KPSTT12}
Anna Kaminska, Alexey I. Popov, Eugeniu Spinu, Adi Tcaciuc, Vladimir G. Troitsky.
``Norm closed operator ideals in Lorentz sequence spaces,''
{\it J. Math. Analysis and Appl.}, 389(1) (2012), pp247-260.

\bibitem[LT77]{LT77}
Joram Lindenstrauss and Lior Tzafriri.
\newblock {\it Classical Banach Spaces I:  Sequence Spaces}
\newblock (1977), ISBN 3-540-08072-4.

\bibitem[OS15]{OS15}
Timur Oikhberg, E. Spinu.
``Subprojective Banach spaces,''
{\it Journal of Mathematical Analysis and Applications} 424:1 (1 April 2015), pp613–635.

%\bibitem[Pe14]{Pe14}
%Petsoulas, Giorgos.
%``A class of $\ell^p$ saturated Banach spaces,''
%{\it Positivity} (June, 2014), Vol. 18 Issue 2, pp287-317.


\bibitem[Pu76]{Pu76}
Lakhpat R. Pujara.
``A uniformly convex Banach space with a Schauder basis which is subsymmetric but not symmetric,''
{\it Proceedings of the American Mathematical Society}
Vol. 54, No. 1 (January, 1976), pp207-210.

\bibitem[Sa04]{Sa04}
B\"unyamin Sar{\i}.
``Envelope functions and asymptotic structures in Banach spaces,''
{\it Studia Mathematica} 164:3 (2004), pp283-306.


%
%\bibitem[SW15]{SW15}
%Gleb Sirotkin and Ben Wallis.
%``Sequence-singular operators'' (2015, preprint). \verb|arXiv:1509.01485v1 [math.FA]|


%\bibitem{Strohmer}
%\newblock T. Strohmer and R. Heath.
%\newblock ``Grassmannian frames with applications to coding and communications.''
%\newblock \emph{Appl. Comp. Harm. Anal.}, 14(3):257-275 (2003).
%
%
\end{thebibliography}
\end{document}

1612.01145.extracted_bib
[AK06]{AK06 Fernando Albiac and Nigel J. Kalton. \newblock {Topics in Banach Space Theory. \newblock Graduate Texts in Mathematics, \newblock Springer Inc. (2006), ISBN 978-0387-28141-4.
1612.01145.extracted_bib
[Al75]{Al75 Zvi Altshuler. ``Uniform convexity in Lorentz sequence spaces,'' {Israel Journal of Mathematics, Volume 20, Nos. 3-4 (1975), pp260-274.
1612.01145.extracted_bib
[ACL73]{ACL73 Zvi Altshuler, Peter G. Casazza, Bor-Luh Lin. ``On symmetric basic sequences in Lorentz sequence spaces,'' {Israel Journal of Mathematics, Volume 15, Issue 2 (June, 1973), pp140-55. % % % %
1612.01145.extracted_bib
[BCFW15]{BCFW15 %Kevin Beanland, Ryan Causey, Daniel Freeman, Ben Wallis. %\newblock ``Classes of operators determined by ordinal indices'' %\newblock (preprint, 2015). \verb|arXiv:1507.06285v2 [math.FA]| % % % % %
1612.01145.extracted_bib
[BF03]{BF03 %\newblock John J. Benedetto and Matthew Fickus. %\newblock ``Finite Normalized Tight Frames.'' %\newblock Advances in Computational Mathematics 18: 357–385, 2003. % %
1612.01145.extracted_bib
[CK12]{CK12 %Peter G. Casazza and Gitta Kutyniok. %\newblock {Finite Frames: Theorey and Applications %\newblock (2012). % %
1612.01145.extracted_bib
[CL74]{CL74 P.G. Casazza, Bor-Luh Lin. ``On symmetric basic sequences in lorentz sequence spaces II,'' {Israel Journal of Mathematics Volume 17, Issue 2 (December 1974), pp191-218.
1612.01145.extracted_bib
[DOSZ11]{DOSZ11 Stephen J. Dilworth, E. Odell, Thomas Schlumprecht, Andras Zs\'ak. ``Renormings and Symmetry Properties of 1-greedy bases,'' {J. Approx Theory 163 (2011), pp1049-1075. %
1612.01145.extracted_bib
[Fab01]{Fab01 %Mari\'an J. Fabian, Petr Habala, Petr H\'ajek, Vicente Montesinos Santaluc\'ia, Jan Pelant, and V\'aclav Zizler. %\newblock {Functional Analysis and Infinite-Dimensional Geometry %\newblock (2001). %
1612.01145.extracted_bib
[En73]{En73 Per Enflo. ``Banach spaces which can be given an equivalent uniformly convex norm,'' {Israel J. Math. 13 (1973), pp281–288.
1612.01145.extracted_bib
[FW08]{FW08 Carlos Finol and Marek Wójtowicz. ``Complemented copies of $\ell_1$ in Banach spaces with an unconditional basis,'' {Journal of Mathematical Analysis and Applications 342:1 (1 June 2008), pp83-88. % %
1612.01145.extracted_bib
[Fe80]{Fe80 %Moshe Feder. %``On subspaces of spaces with an unconditional basis and spaces of operators,'' %{Illinois Journal of Mathematics Volume 24, Number 2 (summer, 1980), pp196-205. % %
1612.01145.extracted_bib
{Casazza %\newblock Peter G. Casazza, Gitta Kutyniok and Friedrich Philipp. %\newblock Introduction to Finite Frame Theory. %\newblock (internet document) %\newblock \begin{verbatimhttp://www.springer.com/cda/content/document
1612.01145.extracted_bib
[Ga68]{Ga68 D.J.H. Garling. ``Symmetric bases of locally convex spaces,'' {Studia Mathematica, Vol. 30, No. 2 (1968), pp163-81.
1612.01145.extracted_bib
[KPSTT12]{KPSTT12 Anna Kaminska, Alexey I. Popov, Eugeniu Spinu, Adi Tcaciuc, Vladimir G. Troitsky. ``Norm closed operator ideals in Lorentz sequence spaces,'' {J. Math. Analysis and Appl., 389(1) (2012), pp247-260.
1612.01145.extracted_bib
[LT77]{LT77 Joram Lindenstrauss and Lior Tzafriri. \newblock {Classical Banach Spaces I: Sequence Spaces \newblock (1977), ISBN 3-540-08072-4.
1612.01145.extracted_bib
[OS15]{OS15 Timur Oikhberg, E. Spinu. ``Subprojective Banach spaces,'' {Journal of Mathematical Analysis and Applications 424:1 (1 April 2015), pp613–635. %
1612.01145.extracted_bib
[Pe14]{Pe14 %Petsoulas, Giorgos. %``A class of $\ell^p$ saturated Banach spaces,'' %{Positivity (June, 2014), Vol. 18 Issue 2, pp287-317.
1612.01145.extracted_bib
[Pu76]{Pu76 Lakhpat R. Pujara. ``A uniformly convex Banach space with a Schauder basis which is subsymmetric but not symmetric,'' {Proceedings of the American Mathematical Society Vol. 54, No. 1 (January, 1976), pp207-210.
1612.01145.extracted_bib
[Sa04]{Sa04 B\"unyamin Sar{\i. ``Envelope functions and asymptotic structures in Banach spaces,'' {Studia Mathematica 164:3 (2004), pp283-306. % %
1612.01145.extracted_bib
[SW15]{SW15 %Gleb Sirotkin and Ben Wallis. %``Sequence-singular operators'' (2015, preprint). \verb|arXiv:1509.01485v1 [math.FA]| %
1612.01145.extracted_bib
{Strohmer %\newblock T. Strohmer and R. Heath. %\newblock ``Grassmannian frames with applications to coding and communications.'' %\newblock Appl. Comp. Harm. Anal., 14(3):257-275 (2003). % %