Shrinking Schauder Frames and their Associated Bases

A frame for a separable infinite dimensional Hilbert space $H$ is a sequence of vectors $(x_{j})_{j=1}^{\infty}$ in $H$ such that there exists constants $0<A\leq B$ so that $A\|x\|^{2}\leq\sum|\langle x,x_{j}\rangle|^{2}\leq B\|x\|^{2}$ for all $x\in H$. If $(x_{j})_{j=1}^{\infty}$ is a frame of $H$ then there exists a possibly different frame $(f_{j})_{j=1}^{\infty}$ of $H$ called a dual frame such that

That is, frames can be used like a basis to give a linear reconstruction formula for vectors in $H$. The difference between frames and bases is that a frame allows for redundancy in that the coefficients given for reconstruction in (1) are not required to be unique.

Frames have been generalized to Banach spaces in various ways such as atomic decompositions [7, 9], framings [6], and Schauder frames [3, 5]. We will focus on Schauder frames in this paper which are a direct generalization of the reconstruction formula given in (1). Let $X$ be a separable infinite dimensional Banach space. A sequence of pairs $(x_{j},f_{j})_{j=1}^{\infty}$ in $X\times X^{*}$ is called a Schauder frame for $X$ if

We make the convention that $x_{j}\not=0$ for $j\in\mathbb{N}$. Though Schauder frames were not explicitly defined until 2008, the first appearance of a Schauder frame, without the name Schauder frame, is in A. Pełczyński’s proof [16] that every space with the bounded approximation property (BAP) is isomorphic to complemented subspace of a space with a basis¹¹1In the same year (1971) Johnson, Rosenthal, and Zippin [10] proved the same result with a completely different method.. In 1987, S. Szarek showed that spaces with the BAP but without bases exist [17]. Indeed, Pełczyński showed that a separable Banach space $X$ has the BAP if and only if it has a Schauder frame (in the above sense) and that, furthermore, there is a space $Z$ with a basis so that the identity on $X$ factors through the identity on $Z$ in a natural way. Formally, if $(x_{i},f_{i})_{i=1}^{\infty}$ is a Schauder frame for $X$ and $Z$ is a Banach space with a Schauder basis $(z_{i})_{i=1}^{\infty}$ then $Z$ is an associated space for $(x_{i},f_{i})_{i=1}^{\infty}$ (and $(z_{i})_{i=1}^{\infty}$ is an associated basis for the frame $(x_{i},f_{i})_{i=1}^{\infty}$) if the maps $T:X\to Z$ (analysis operator) and $S:Z\to X$ (synthesis operator) defined by $T(\sum_{i=1}^{\infty}f_{i}(x)x_{i})=\sum_{i=1}^{\infty}f_{i}(x)z_{i}$ and $S(\sum_{i=1}^{\infty}a_{i}z_{i})=\sum_{i=1}^{\infty}a_{i}x_{i}$ are bounded.

In the case that both $(x_{i})_{i=1}^{\infty}$ and $(f_{i})_{i=1}^{\infty}$ are frames for a Hilbert space $H$, then the associated space $Z$ can be chosen to be $\ell_{2}$ and the associated basis $(z_{i})_{i=1}^{\infty}$ can be chosen to be the unit vector basis for $\ell_{2}$. This is of fundamental importance in frame theory as well as in applications such as signal processing. Indeed, given some vector $x\in H$, the analysis operator maps $x$ to $T(x)=(\langle f_{i},x\rangle)_{i=1}^{\infty}\in\ell_{2}$. One can then apply filters to the sequence of frame coefficients $(\langle f_{i},x\rangle)_{i=1}^{\infty}$ to obtain a sequence $(b_{i})_{i=1}^{\infty}\in\ell_{2}$. Applying the synthesis operator then gives a vector $S((b_{i})_{i=1}^{\infty})=\sum_{i=1}^{\infty}b_{i}x_{i}$ which is an approximation of $x$ but is improved in some way such as being compressed or having noise or artifacts removed.

If one wishes to use similar techniques for a Schauder frame $(x_{i},f_{i})_{i=1}^{\infty}$ then it is advantageous to construct the associated basis $(z_{i})_{i=1}^{\infty}$ to be as nice as possible. That is, if $(x_{i},f_{i})_{i=1}^{\infty}$ has some desirable property such as being unconditional, shrinking, or boundedly complete then one would like $(z_{i})_{i=1}^{\infty}$ to share the property as well. If $(x_{i},f_{i})_{i=1}^{\infty}$ is unconditional, then it is straightforward to construct an unconditional associated basis. In [1], the authors of the current paper and R. Liu prove that if $(x_{i},f_{i})_{i=1}^{\infty}$ is shrinking then it has a shrinking associated basis. However, the construction in [1] is relatively difficult and involves the method of bounds on branches of weakly null trees developed by E. Odell and Th. Schlumprecht [15] [8]. One of the main goals of this paper is to give a more direct and much simpler construction of a shrinking associated basis, which we state in the following theorem.

Given a Schuader frame $(x_{i},f_{i})_{i=1}^{\infty}$ with $x_{i}\neq 0$ for all $i\in\mathbb{N}$, the most natural associated space is now referred to as the minimal associated space [5][12] and is defined as follows. Denote by $(z_{i})_{i=1}^{\infty}$ the unit vector basis for $c_{00}$ and for $(a_{i})\in c_{00}$ consider the norm

The minimal associated space $Z_{min}$ is defined to be the completion of $c_{00}$ under the above norm and the basis $(z_{i})_{i=1}^{\infty}$ is called the minimal associated basis. A Schauder frame $(x_{i},f_{i})_{i=1}^{\infty}$ may have many non-equivalent associated bases. However, the basis $(z_{i})_{i=1}^{\infty}$ defined in (5) is minimal in the sense that if $(y_{i})_{i=1}^{\infty}$ is any associated basis for $(x_{i},f_{i})_{i=1}^{\infty}$ then $(y_{i})_{i=1}^{\infty}$ dominates $(z_{i})_{i=1}^{\infty}$. That is, there exists a constant $K>0$ so that $\|\sum_{i}a_{i}z_{i}\|\leq K\|\sum_{i}a_{i}y_{i}\|$ for all $(a_{i})\in c_{00}$ [12].

We now consider the problem of determining if a shrinking Schauder frame has a minimal shrinking associated basis. That is, if $(x_{i},f_{i})_{i=1}^{\infty}$ is a shrinking Schauder frame, then when does there exist a shrinking associated basis $(w_{i})_{i=1}^{\infty}$ of $(x_{i},f_{i})_{i=1}^{\infty}$ such that if $(y_{i})_{i=1}^{\infty}$ is any shrinking associated basis of $(x_{i},f_{i})_{i=1}^{\infty}$ then $(y_{i})_{i=1}^{\infty}$ dominates $(w_{i})_{i=1}^{\infty}$? In Section 4 we prove that a Schauder frame has a minimal shrinking associated basis if and only if the minimal associated basis defined in (5) is shrinking. Our construction of a shrinking associated basis in Theorem 1.1 is defined solely in terms of the shrinking Schauder frame $(x_{i},f_{i})_{i=1}^{\infty}$ and some sequence of natural numbers $(N_{k})_{k=1}^{\infty}\in[\mathbb{N}]^{\omega}$. In Section 4 we prove that if $(x_{i},f_{i})_{i=1}^{\infty}$ is a shrinking Schauder frame and $(y_{i})_{i=1}^{\infty}$ is any shrinking associated basis then there exists $(N_{k})_{k=1}^{\infty}\in[\mathbb{N}]^{\omega}$ such that the resulting shrinking associated basis $(z_{i})_{i=1}^{\infty}$ from our construction is dominated by $(y_{i})_{i=1}^{\infty}$. In other words, Theorem 1.1 produces a set of shrinking associated bases such that every shrinking associated basis of $(x_{i},f_{i})_{i=1}^{\infty}$ dominates some basis in that set. Furthermore, we prove that if the minimal associated basis is not shrinking then for every shrinking associated basis $(y_{i})_{i=1}^{\infty}$ there exists uncountably many mutually incomparable shrinking associated bases which are all dominated by $(y_{i})_{i=1}^{\infty}$. Hence, except for the trivial case where the minimal associated basis is shrinking, we have that the collection of shrinking associated bases will have a very rich lattice structure under the domination partial order.

In the final section we make some observations about how this work relates to now classical results about the BAP and give an alternative proof of the theorem of Johnson, Rosenthal, and Zippin that for a Banach space $X$ with separable dual, $X^{*}$ has the BAP if and only if $X$ is isomorphic to a complemented subspace of a Banach space with a shrinking basis.

A sequence of vectors $(x_{i})_{i=1}^{\infty}$ in a separable Banach space $X$ is called a Schauder basis if for all $x\in X$ there exists a unique sequence of scalars $(a_{i})_{i=1}^{\infty}$ such that $x=\sum_{i=1}^{\infty}a_{i}x_{i}$. If $X$ is a Banach space with dual $X^{*}$ then a Schauder basis $(x_{i})_{i=1}^{\infty}$ is called shrinking if the biorthogonal functionals $(x_{i}^{*})_{i=1}^{\infty}$ form a Schauder basis for $X^{*}$. In particular, a Banach space with a shrinking basis necessarily has a separable dual with a basis. There are, however, Banach spaces with bases whose duals are separable but fail the approximation property [11, Theorem 1.e.7.(b)]. Naturally, a Schauder frame $(x_{i},f_{i})_{i=1}^{\infty}$ for $X$ is shrinking if and only if $(f_{i},x_{i})_{i=1}^{\infty}$ is a Schauder frame for $X^{*}$. Using the terminology atomic decomposition instead of Schauder frame, Carando and Lassalle [2] give the following useful characterization of shrinking Schauder frames which is analogous to James’ well-known characterization for Schauder bases.

We sketch a short proof for completeness.

Let $X$ be a Banach space with a Schauder frame $(x_{i},f_{i})_{i=1}^{\infty}$ and let $Z$ be a Banach space with a Schauder basis $(z_{i})_{i=1}^{\infty}$. Recall that $Z$ is said to be an associated space of the Schauder frame $(x_{i},f_{i})_{i=1}^{\infty}$ and $(z_{i})_{i=1}^{\infty}$ is said to an associated basis if the maps $T:X\rightarrow Z$ and $S:Z\rightarrow X$ are bounded where $T(x)=\sum_{i=1}^{\infty}f_{i}(x)x_{i}$ for all $x\in X$ and $S(\sum_{i=1}^{\infty}a_{i}z_{i})=\sum_{i=1}^{\infty}a_{i}x_{i}$ for all $\sum_{i=1}^{\infty}a_{i}z_{i}\in Z$. It follows immediately that if a Schauder frame has a shrinking associated basis then the Schauder frame must be shrinking. In [1], the authors of the current paper and R. Liu prove a more general and technical theorem which implies that every shrinking Schauder frame has a shrinking associated basis. Unfortunately, the argument does not provide an explicit construction of the associated basis and the proofs are relatively difficult. In Section 3 we give an explicit method which will give a shrinking associated basis for any shrinking Schauder frame. Before proceeding we show that the minimal associated basis for a shrinking Schauder frame need not be shrinking.

The above example, although simple, is rather instructive in that it reveals that redundancy in a frame can make the minimal associated basis not shrinking.

A sequence $(y_{i})$ in a Banach space is called $\alpha$-$\ell_{1}^{+}$ for some $\alpha>0$ if $\|\sum a_{i}y_{i}\|\geqslant\alpha\sum a_{i}$ whenever $(a_{i})$ is a summable sequence of non-negative scalars. We conclude this section by recalling the following useful and well known characterization of shrinking bases.

We start by setting some notation. Let $(x_{i},f_{i})_{i=1}^{\infty}$ be a shrinking Schauder frame for a Banach space $X$. Let $Z$ be an associated space and $(z_{i})_{i=1}^{\infty}$ be an associated basis. The analysis operator is the map $T:X\rightarrow Z$ given by $T(x)=\sum_{i=1}^{\infty}f_{i}(x)z_{i}$ for all $x\in X$. The synthesis operator is the map $S:Z\rightarrow X$ given by $S(\sum_{i=1}^{\infty}a_{i}z_{i})=\sum_{i=1}^{\infty}a_{i}x_{i}$ for all $\sum_{i=1}^{\infty}a_{i}z_{i}\in Z$. For $m<n$, we use the following notation when we wish to use partial sums.

1. 1.

   $S_{[m,n]}(\sum_{i}a_{i}z_{i})=\sum_{i=m}^{n}a_{i}x_{i}$ for all $\sum a_{i}z_{i}\in Z$,

2. 2.

   $R_{[m,n]}(\sum_{i}a_{i}z_{i})=\sum_{i=m}^{n}a_{i}z_{i}$ for all $\sum a_{i}z_{i}\in Z$,

3. 3.

   $P_{[m,n]}(x)=\sum_{i=m}^{n}f_{i}(x)x_{i}$.

It follows from the uniform boundedness principle that $\sup_{m\leq n}\|S_{[m,n]}\|$, $\sup_{m\leq n}\|R_{[m,n]}\|$, and $\sup_{m\leq n}\|P_{[m,n]}\|$ are all finite. The value $\sup_{m\leq n}\|R_{[m,n]}\|$ is the basis constant of $(z_{i})_{i=1}^{\infty}$ and the value $\sup_{m\leq n}\|P_{[m,n]}\|$ is called the frame constant of $(x_{i},f_{i})_{i=1}^{\infty}$.

The following proposition is contained in [1] and we include the short proof for completeness.

We now define a norm on $c_{00}$ which we will later prove gives a shrinking associated basis.

Proposition 3.1 gives a condition satisfied by each shrinking Schauder frame. The idea behind the definition of the norm above is to force the associated space to satisfy some version of this condition. The goal then is to show that satisfying this condition is sufficient to establish that the associated basis is shrinking.