UPPER AND LOWER ESTIMATES FOR SCHAUDER FRAMES AND ATOMIC DECOMPOSITIONS. K. BEANLAND, D. FREEMAN, AND R. LIU. i=1. i=1

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1 UPPER AND LOWER ESTIMATES FOR SCHAUDER FRAMES AND ATOMIC DECOMPOSITIONS. K. BEANLAND, D. FREEMAN, AND R. LIU Abstract. We prove that a Schauder frame for any separable Banach space is shrinking if and only if it has an associated space with a shrinking basis, and that a Schauder frame for any separable Banach space is shrinking and boundedly complete if and only if it has a reflexive associated space. To obtain these results, we prove that the upper and lower estimate theorems for finite dimensional decompositions of Banach spaces can be extended and modified to Schauder frames. We show as well that if a separable infinite dimensional Banach space has a Schauder frame, then it also has a Schauder frame which is not shrinking. 1. Introduction Frames for Hilbert spaces were introduced by Duffin and Schaeffer in 1952 [DS] to address some questions in non-harmonic Fourier series. However, the current popularity of frames is largely due to their successful application to signal processing which was initiated by Daubechies, Grossmann, and Meyer in 1986 [DGM]. A frame for an infinite dimensional separable Hilbert space H is a sequence of vectors (x i ) H for which there exists constants 0 A B such that for any x H, (1) A x 2 x, x i 2 B x 2. If A = B = 1, then (x i ) is called a Parseval frame. Given any frame (x i ) for a Hilbert space H, there exists a frame (f i ) for H, called an alternate dual frame, such that for all x H, (2) x = x, f i x i. The equality in (2) allows the reconstruction of any vector x in the Hilbert space from the sequence of coefficients ( x, f i ). The standard method to construct such a frame (f i ) is to take f i = S 1 x i for all i N, where S is the positive, self-adjoint invertible operator on H Date: February 6, Mathematics Subject Classification. Primary 46B20, Secondary 41A65. Key words and phrases. Banach spaces; Frames; Schauder Frames; Atomic Decompositions. The research of the second author was supported by a grant from the National Science Foundation. The third author is supported by the National Nature Science Foundation of China (No ), the Fundamental Research Fund for the Central Universities of China, and the Tianjin Science & Technology Fund. 1

2 defined by Sx = x, x i x i for all x H. The operator S is called the frame operator and the frame (S 1 x i ) is called the canonical dual frame of (x i ). In their AMS memoir [HL], Han and Larson initiated studying the dilation viewpoint of frames. That is, analyzing frames as orthogonal projections of Riesz bases, where a Riesz basis is an unconditional basis for a Hilbert space. To start this approach, they proved the following theorem. Theorem 1.1 ([HL]). If (x i ) is a frame for a Hilbert space H, then there exists a larger Hilbert space Z H and a Riesz basis (z i ) for Z such that P X z i = x i for all i N, where P X is orthogonal projection onto X. Furthermore, if (x i ) is Parseval, then (z i ) can be taken to be an ortho-normal basis. Recently, Theorem 1.1 was extended to operator-valued measures and bounded linear maps [HLLL]. Recently as well, a continuous version of Theorem 1.1 was given for vector bundles and Riemannian manifolds [FPWW]. The concept of a frame was extended to Banach spaces in 1988 by Feichtinger and Gröchenig [FG] (and more generally in [G]) through the introduction of atomic decompositions. The main goal of [G] was to obtain for Banach spaces the unique association of a vector with the natural set of frame coefficients. In 2008, Schauder frames for Banach space were developed [CDOSZ] with the goal of creating a procedure to represent vectors using quantized coefficients. A Schauder frame essentially takes, as its definition, an extension of the equation (2) to Banach spaces. Definition 1.2. Let X be an infinite dimensional separable Banach space. A sequence (x i, f i ) X X is called a Schauder frame for X if x = f i(x)x i for all x X. In particular, if (x i ) and (f i ) are frames for a Hilbert space H, then (f i ) is an alternate dual frame for (x i ) if and only if (x i, f i ) is a Schauder frame for H. As noted in [CDOSZ], a separable Banach space has a Schauder frame if and only if it has the bounded approximation property. By the uniform boundedness principle, for any Schauder frame (x i, f i ) of a Banach space X, there exists a constant C 1 such that sup n m n i=m f i(x)x i C x for all x X. The least such value C is called the frame constant of (x i, f i ). A Schauder frame (x i, f i ) is called unconditional if the series x = f i(x)x i converges unconditionally for all x X. The following definitions allow the dilation viewpoint of Han and Larson to be extended to Schauder frames. Definition 1.3. Let (x i, f i ) be a frame for a Banach space X and let Z be a Banach space with basis (z i ) and coordinate functionals (z i ). We call Z an associated space to (x i, f i ) and (z i ) an associated basis if the operators T : X Z and S : Z X are bounded, where, T (x) = T ( f i (x)x i ) = f i (x)z i for all x X and S(z) = S( z i (z)z i ) = z i (z)x i for all z Z. Essentially, Theorem 1.1 states that a frame for a Hilbert space has an associated basis which is a Riesz basis for a Hilbert space. Furthermore, the proof in [HL] actually involves constructing the operators T and S given in Definition 1.3. In [CHL], it is shown that every 2

3 Schauder frame has an associated space, which is referred to as the minimal associated space in [L]. Being able to dilate a Parseval frame for a Hilbert space to an ortho-normal basis is very useful in understanding and working with frames for Hilbert spaces. The minimal associated basis may be used similarly for studying Schauder frames. However, if a Schauder frame has some useful property, then an associated basis with a corresponding property should be used to study the Schauder frame. To take full advantage of this approach, we need to characterize Schauder frame properties in terms of associated bases. This would allow the large literature on Schauder basis properties to be applied to Schauder frames. If (x i, f i ) is a Schauder frame, then the reconstruction operator, S, for the minimal associated space contains c 0 in its kernel if and only if a finite number of vectors can be removed from (x i ) to make it a Schauder basis [LZ]. This implies in particular that except in trivial cases, the minimal associated basis will not be boundedly complete and the minimal associated space will not be reflexive. Thus, to work with a reflexive associated space, we will need to consider a new method of constructing associated spaces. A Banach space with a basis is reflexive if and only if the basis is shrinking and boundedly complete. In order to characterize when a Schauder frame has a reflexive associated space, it is then natural to consider a generalization of the properties shrinking and boundedly complete from the context of bases to that of Schauder frames. Definition 1.4. Given a Schauder frame (x i, f i ) X X, let T n : X X be the operator T n (x) = i n f i(x)x i. The frame (x i, f i ) is called shrinking if x T n 0 for all x X. The frame (x i, f i ) is called boundedly complete if x (f i )x i converges in norm to an element of X for all x X. As noted in [CL], if (x i ) is a Schauder basis and (x i ) are the biorthogonal functionals of (x i ), then the frame (x i, x i ) is shrinking if and only if the basis (x i ) is shrinking, and the frame (x i, x i ) is boundedly complete if and only if the basis (x i ) is boundedly complete. Thus the definition of a frame being shrinking or boundedly complete is consistent with that of a basis. In [L], the frame properties shrinking and boundedly complete are called pre-shrinking and pre-boundedly complete. It is not difficult to see that if a Schauder frame has a shrinking associated basis, then the frame must be shrinking as well, and that if a Schauder frame has a boundedly complete associated basis, then the frame must be boundedly complete. In [L], the minimal and maximal associated spaces are defined and it is proven that if a frame is shrinking and satisfies some strong local conditions, then the minimal associated basis is shrinking and that if a frame is boundedly complete and satisfies some strong local conditions then the maximal associated basis is boundedly complete. The advantage of these theorems is that explicit associated bases are constructed with the desired properties of shrinking or boudendly complete. On the other hand, not every Schauder frame with a shrinking or boundedly complete associated basis satisfies the local conditions given in [L]. Thus, it is not a complete characterization of which frames have a shrinking associated basis or of which frames have a boundedly complete associated basis. In the following theorem, we give a complete characterization. 3

4 Theorem 1.5. Let (x i, f i ) be a Schauder frame for a Banach space X. Then (x i, f i ) is shrinking if and only if (x i, f i ) has a shrinking associated basis. Furthermore, (x i, f i ) is shrinking and boundedly complete if and only if (x i, f i ) has a reflexive associated space. The properties of shrinking and boundedly complete do not immediately give us much structure to directly create an associated space. To obtain Theorem 1.5, we will prove a stronger result, involving upper and lower estimates. In [OSZ2], it is shown that every Banach space with separable dual satisfies certain upper estimates and that every separable reflexive Banach space satisfies certain upper and lower estimates. As we will be using these estimates, we will define them precisely in Section 3. One of the main results of [OSZ2] is that every separable reflexive Banach space embeds into a reflexive Banach space with a finite dimensional decomposition (FDD) satisfying the same upper and lower estimates, and one of the main results of [FOSZ] is that every Banach space with separable dual embeds into a Banach space with a shrinking FDD satisfying the same upper estimate. We extend the techniques developed in [OSZ1] to Schauder frames, and prove the following theorems which are extensions of the results from [OSZ2] and [FOSZ] and will be used to prove Theorem 1.5. Theorem 1.6. Let X be a Banach space with a shrinking Schauder frame (x i, f i ). Let (v i ) be a normalized, 1-unconditional, block stable, right dominant, and shrinking basic sequence. If X satisfies subsequential (v i ) upper tree estimates, then there exists (n i ), (K i ) [N] ω and an associated space Z with a shrinking basis (z i ) such that the FDD (span j [ni,n i+1 )z i ) satisfies subsequential (v Ki ) upper block estimates. Theorem 1.7. Let X be a Banach space with a shrinking and boundedly complete Schauder frame (x i, f i ). Let (u i ) be a normalized, 1-unconditional, block stable, right dominant, and shrinking basic sequence, and let (v i ) be a normalized, 1-unconditional, block stable, left dominant, and boundedly complete basic sequence such that (u i ) dominates (v i ). Then X satisfies subsequential (u i ) upper tree estimates and subsequential (v i ) lower tree estimates, if and only if there exists (n i ), (K i ) [N] ω and a reflexive associated space Z with associated basis (z i ) such that the FDD (span j [ni,n i+1 )z j ) satisfies subsequential (u Ki ) upper block estimates and subsequential (v Ki ) lower block estimates. Using the theory of trees and branches in Banach spaces developed by Odell and Schlumprecht, we have that Theorem 1.5 follows immediately from Theorems 1.6 and 1.7. The basic idea is that every Banach space with separable dual satisfies subsequential (v i ) upper tree estimates for some normalized, 1-unconditional, block stable, right dominant, and shrinking basic sequence (v i ). If X is a Banach space with a shrinking Schauder frame, then it is known that X must have separable dual and hence we may apply Theorem 1.6 to obtain a shrinking associated basis to the shrinking Schauder frame. Similarly, every separable reflexive Banach space satisfies subsequential (u i ) upper tree estimates and subsequential (v i ) lower tree estimates for some basic sequences (u i ) and (v i ) satisfying the hypotheses of Theorem 1.7. If X is a Banach space with a shrinking and boundedly complete Schauder frame, then it is known that X must be reflexive and hence we may apply Theorem 1.7 to obtain a reflexive associated space for the Schauder frame. Because Theorems 1.5, 1.6, and 1.7 are our main 4

5 theorems, at the end of Section 3 we will prove (assuming some well known properties about Tsirelson spaces) how Theorem 1.5 follows from Theorems 1.6 and 1.7. One of the main goals in proving the upper and lower estimates theorems in [OSZ1] and [FOSZ] was to obtain embedding results that preserve quantitative bounds on the Szlenk index. In [Z], Zippin proves that every Banach space with separable dual embeds into a Banach space with a shrinking basis and that every separable reflexive Banach space embeds into a Banach space with a shrinking and boundedly complete basis. On the other hand, there does not exist a single Banach space Z with a shrinking basis such that every reflexive Banach space embeds into Z [Sz]. To prove this result, Szlenk created an ordinal index Sz on the set of separable Banach spaces which satisfies (1) Sz(X) is countable if and only if X has separable dual, (2) if X embeds into Y then Sz(X) Sz(Y ), and (3) for every countable ordinal α there exists a Banach space with separable dual X such that Sz(X) > α. Thus, if Z is a separable Banach space such that every Banach space with separable dual embeds into Z then Sz(Z) is uncountable and hence Z does not have separable dual. There have been many further results in this direction, for example Bourgain proved that if Z is a separable Banach space such that every separable reflexive Banach space embeds into Z then also every separable Banach space embeds into Z [B]. Bourgain then asked if there exists a separable reflexive Banach space Z such that every separable uniformly convex Banach space embeds into Z. In [OS2] Odell and Schlumprecht construct such a space answering Bourgain s question. Inspired by this paper, Pe lczyński asked if for every countable ordinal α there exists a reflexive Banach space Z such that every separable reflexive Banach space X with max(sz(x), Sz(X )) α embeds into Z. This was proved in [OSZ2], where it was shown that for every countable ordinal α there exists a separable reflexive Banach space Z with max{sz(z), Sz(Z )} ω αω+1 such that every separable reflexive Banach space X with max{sz(x), Sz(X )} ω αω embeds into Z. In [FOSZ] it was proven that for every countable ordinal α there exists a separable Banach space Z with Sz(Z) ω αω+1 such that every separable Banach space X with Sz(X) ω αω embeds into Z. Causey recently extended this characterization to more ordinals using Schreier spaces [C1] and generalized Baernstein spaces [C2]. The approach to proving these embedding results for the Szlenk index was to first prove an equivalence between certain bounds on the Szlenk index of a Banach space and certain upper estimates, and then to prove that one can embed into a Banach space with an FDD which preserved the upper estimates. In particular, it is proven in [OSZ2] that a Banach space X with separable dual has Szlenk index at most ω αω if and only if X satisfies subsequential T α,c -upper tree estimates for some constant 0 < c < 1, where T α,c is the Tsirelson space of order α and constant c. Using this equivalence between upper Tsirelson estimates and bounds on the Szlenk index, we obtain the following corollaries as immediate applications of Theorem 1.6 and Theorem 1.7. Corollary 1.8. Let (x i, f i ) be a shrinking Schauder frame for a Banach space X and let α be a countable ordinal. Then, the following are equivalent. (a) X has Szlenk index at most ω αω. (b) X satisfies subsequential T α,c -upper tree estimates for some constant 0 < c < 1, where T α,c is the Tsirelson space of order α and constant c. 5

6 (c) (x i, f i ) has an associated shrinking basis (z i ) such that there exists (n i ), (K i ) [N] ω and 0 < c < 1 so that the FDD (span j [ni,n i+1 )z i ) satisfies subsequential (t Ki ) upper block estimates, where (t i ) is the unit vector basis for T α,c. (d) (x i, f i ) has an associated Banach space with a shrinking basis and Szlenk index at most ω αω. Corollary 1.9. Let (x i, f i ) be a shrinking and boundedly complete Schauder frame for a Banach space X and let α be a countable ordinal. Then, the following are equivalent. (a) X and X both have Szlenk index at most ω αω. (b) X satisfies subsequential T α,c -upper tree estimates and subsequential T α,c-lower tree estimates for some constant 0 < c < 1. (c) (x i, f i ) has an associated shrinking and boundedly complete basis (z i ) such that there exists (n i ), (K i ) [N] ω and 0 < c < 1 so that the FDD (span j [ni,n i+1 )z i ) satisfies subsequential (t Ki ) upper block estimates and subsequential (t K i ) lower block estimates, where (t i ) is the unit vector basis for T α,c. (d) (x i, f i ) has an associated reflexive Banach space Z such that Z and Z both have Szlenk index at most ω αω. Both Schauder frames and atomic decompositions are natural extensions of frame theory into the study of Banach space. These two concepts are directly related, and some papers in the area such as [CHL], [CL], and [CLS] are stated in terms of atomic decompositions, while others such as [CDOSZ], [L], and [LZ] are stated in terms of Schauder frames. Definition Let X be a Banach space and Z be a Banach sequence space. We say that a sequence of pairs (x i, f i ) X X is an atomic decomposition of X with respect to Z if there exists positive constants A and B such that for all x X: (a) (f i (x i )) Z, (b) A x (f i (x i )) Z B x, (c) x = f i(x)x i. If the unit vectors in the Banach space Z given in Definition 1.10 form a basis for Z, then an atomic decomposition is simply a Schauder frame with a specified associated space Z. We choose to use the terminology of Schauder frames for this paper instead of atomic decomposition as, to us, an associated space is an object which is useful for studying the Schauder frame but is external to the space X and frame (x i, f i ). Our goals are essentially, to construct nice associated spaces, given a particular Schauder frame. However, our theorems can be stated in terms of atomic decompositions. In particular, Theorem 1.5 can be stated as the following. Theorem Let X be a Banach space and Z be a Banach sequence space whose unit vectors form a basis for Z. Let (x i, f i ) be an atomic decomposition of X with respect to Z. Then (x i, f i ) is shrinking if and only if there exists a Banach sequence space Z whose unit vectors form a shrinking basis for Z such that (x i, f i ) is an atomic decomposition of X with respect to Z. Furthermore, (x i, f i ) is shrinking and boundedly complete if and only if there exists a 6

7 reflexive Banach sequence space Z whose unit vectors form a basis for Z such that (x i, f i ) is an atomic decomposition of X with respect to Z. The third author would like to express his gratitude to Edward Odell for inviting him to visit the University of Texas at Austin in the fall of 2010 and spring of Shrinking and boundedly complete Schauder Frames It is well known that a basis (x i ) for a Banach space X is shrinking if and only if the biorthogonal functionals (x i ) form a boundedly complete basis for X. The following theorem extends this useful characterization to Schauder frames. Theorem 2.1. [CL, Proposition 2.3][L, Proposition 4.8] Let X be a Banach space with a Schauder frame (x i, f i ) X X. The frame (x i, f i ) is shrinking if and only if (f i, x i ) is a boundedly complete Schauder frame for X. It is a classic and fundamental result of James that a basis for a Banach space is both shrinking and boundedly complete, if and only if the Banach space is reflexive. The following theorem shows that one side of James characterization holds for frames. Theorem 2.2. [CL, Proposition 2.4][L, Proposition 4.9] If (x i, f i ) is a shrinking and boundedly complete Schauder frame of a Banach space X, then X is reflexive. It was left as an open question in [CL] whether the converse of Theorem 2.2 holds. The following theorem shows that this is false for any Banach space X, and is evidence of how general Schauder frames can exhibit fairly unintuitive structure. Theorem 2.3. Let X be a Banach space which admits a Schauder frame (i.e. has the bounded approximation property), then X has a Schauder frame which is not shrinking. Proof. Let (x i, f i ) be a Schauder frame for X. If (x i, f i ) is not shrinking, then we are done. Thus we assume that (x i, f i ) is shrinking. Fix x X such that x 0, and choose (y i ) S X such that y i w 0. For all n N, we define elements (x 3n 2, f 3n 2), (x 3n 1, f 3n 1), (x 3n, f 3n) X X, in the following way: x 3n 2 = x n x 3n 1 = x x 3n = x f 3n 2 = f n f3n 1 = yn f 3n = yn As yi w 0, it is not difficult to see that ( x i, f i ) is a frame of X. However, ( x i, f i ) is not shrinking. Indeed, let x X such that x (x) = 1. As (x i, f i ) is shrinking, there exists N 0 N such that i=m x (x i )f i (y) 1 y for all y X and M N 4 0. Let M N 0 and choose y B X such that ym (y) > 3. We now have the following estimate, 4 x T 3M x T 3M (y) = x (x)ym(y) + f i (y)x (x i ) > = 1 2 i=m+1 Thus we have that x T N 0, and hence ( x i, f i ) is not shrinking. 7

8 As a Schauder frame must be shrinking in order to have a shrinking associated basis, Theorem 2.3 implies that not every Schauder frame for a reflexive Banach space has a reflexive associated space. Definition 2.4. If (x i, f i ) is a Schauder frame for a Banach space X, and (z i ) is an associated basis then (x i, f i ) is called strongly shrinking relative to (z i ) if x S n 0 where S n : Z X is defined by S n (z) = i=n z i (z)x i. for all x X It is clear that if a Schauder frame is strongly shrinking relative to some associated basis, then the Schauder frame must be shrinking. Also, if a Schauder frame has a shrinking associated basis, then the frame is strongly shrinking relative to the basis. In [CL], examples of shrinking Schauder frames are given which are not strongly shrinking relative to some given associated spaces. However, we will prove later that for any given shrinking Schauder frame, there exists an associated basis such that the frame is strongly shrinking relative to the basis. Before proving this, we state the following theorem which illustrates why the concept of strongly shrinking will be important to us and allows us to use frames in duality arguments. Theorem 2.5. [CL, Lemma 1.7, Theorem 1.8] If (x i, f i ) is a Schauder frame for a Banach space X, and (z i ) is an associated basis for (x i, f i ) then (zi ) is an associated basis for (f i, x i ), if and only if (x i, f i ) is strongly shrinking relative to (z i ). Furthermore, given operators T : X Z and S : Z X defined by T (x) = f i (x)z i for all x X and S(z) = zi (z)x i for all z Z, if (x i, f i ) is strongly shrinking relative to (z i ), then S : X [zi ] and T : [zi ] X are given by S (x ) = x (x i )zi for all x X and T (z ) = z (z i )f i for all z [zi ]. Applying Theorem 2.5 to reflexive Banach spaces gives the following corollary. Corollary 2.6. If (x i, f i ) is a shrinking frame for a reflexive Banach space X and (z i ) is an associated basis such that (x i, f i ) is strongly shrinking relative to (z i ) then (f i, x i ) is strongly shrinking relative to (z i ). Before proceeding further, we need some stability lemmas. Note that if (z i ) is a basis for a Banach space Z, with projection operators P (n,k) : Z Z given by P (n,k) ( a i z i ) = i (n,k) a iz i, then P (1,k) P (n, ) = 0 and P (n, ) P (1,k) = 0 for all k < n. The analogous property fails when working with frames. However, the following lemmas will essentially allow us to obtain this property within some given ε > 0 if n is chosen sufficiently larger than k. Lemma 2.7. Let (x i, f i ) be a Schauder frame for a Banach space X. Then for all ε > 0 and k N, there exists N N such that N > k and n n 0 sup f i ( f j (x)x j )x i ε x for all x X. n m N>k n 0 m 0 i=m 0 8

9 Proof. As (x i, f i ) is a Schauder frame, for each 1 l k with f l 0, there exists N l > k such that sup n m Nl n i=m f i(x l )x i < ε/(k f l ). Let N = max 1 l k N l. We now obtain the following estimate for n m N > k n 0 m 0 and x X. n n 0 n f j (x)f i (x j )x i k max f l (x)f i (x l )x i as k n0 1 l k i=m 0 i=m n fl k max f i (x l )x i x ε x as n m N l. 1 l k i=m In terms of operators, Lemma 2.7 can be stated as for all k N, if (x i, f i ) is a Schauder frame then lim N T N (Id X T k ) = 0, where T n : X X is given by T n (x) = i=n f i(x)x i for all n N. We now prove that for all k N, if (x i, f i ) is a shrinking Schauder frame then lim N (Id X T k ) T N = 0. The frame given in the proof of Theorem 2.3 shows that we cannot drop the condition of shrinking. Lemma 2.8. Let (x i, f i ) be a shrinking Schauder frame for a Banach space X. Then for all ε > 0 and k N, there exists N N such that N > k and n 0 n sup f i ( f j (x)x j )x i ε x for all x X. n m N>k n 0 m 0 i=m 0 Proof. By Theorem 2.1, (f i, x i ) is a Schauder frame for X. Thus for each 1 l k with x l 0, there exists N l > k such that sup n m Nl n f l(x j )f j < ε/(k x l ). Let N = max 1 l k N l. We now obtain the following estimate for n m N > k n 0 m 0 and x X. n 0 n n f i ( f j (x)x j )x i k sup f l ( f j (x)x j )x l as k n0 i=m 0 1 l k n = k sup f j (x)f l (x j ) x l 1 l k k sup 1 l k n x xl f l (x j )f j ε x as n m N l. Our method for proving that every shrinking frame has a shrinking associated basis is to first prove that every shrinking frame is strongly shrinking with respect to some associated basis, and then renorm that associated basis to make it shrinking. The following theorem is thus our first major step. 9

10 Theorem 2.9. Let (x i, f i ) be a shrinking Schauder frame for a Banach space X. Then (x i, f i ) has an associated basis (z i ) such that (x i, f i ) is strongly shrinking relative to (z i ). Proof. We repeatedly apply Lemma 2.8 to obtain a subsequence (N k ) k=1 of N such that for all k N, k n (3) sup f i ( f j (x)x j )x i 2 2k x for all x X. n m N k We assume without loss of generality that x i 0 for all i N. We denote the unit vector basis of c 00 by (z i ), and define the following norm, Z for all (a i ) c 00. (4) Z n a i z i = max a i x i max 2 k k n. f i ( a j x j )x i n m k N;n m N k i=m It follows easily that (z i ) is a bimonotone basic sequence, and thus (z i ) is a bimonotone basis for the completion of c 00 under Z, which we denote by Z. We first prove that (z i ) is an associated basis for (x i, f i ). Let C be the frame constant of (x i, f i ). That is, max n m n i=m f i(x)x i C x for all x X. By (3) and (4), the operator T : X Z, defined by T (x) = f i (x)z i for all x X, is bounded and T C. We have that n i=m a iz i Z n i=m a ix i, and hence the operator S : Z X defined by S(z) = zi (z)x i is bounded and S = 1. Thus we have that (z i ) is an associated basis for (x i, f i ). We now prove that (x i, f i ) is strongly shrinking relative to (z i ). Let ε > 0 and x B X. As (x i, f i ) is shrinking, we may choose k N such that 2 k < ε/2 and j=k+1 x (x j )f j < ε/2. We obtain the following estimate for any N N k and z = a i z i Z. x ( a i x i ) = x (x j )f j ( a i x i ) as (f i, x i ) is a frame for X i=n = j=1 i=n k x (x j )f j ( a i x i ) + j=1 x i=n j=k+1 k f j ( a i x i )x j + j=1 i=n x 2 k z Z + < ε/2 z Z + ε/2 z Z j=k+1 x (x j )f j ( a i x i ) j=k+1 x (x j )f j z Z i=n x (x j )f j z Z as by (4) as N N k a i x i z Z We thus have that for all x X and ε > 0, that there exists M N such that x ( i=n z i (z)x i ) < ε for all N M and z B Z. Hence, (x i, f i ) is strongly shrinking relative to (z i ). 10 i=n

11 The following lemmas incorporate an associated basis into the tail and initial segment estimates of Lemma 2.7 and Lemma 2.8. Lemma Let X be a Banach space with a shrinking Schauder frame (x i, f i ). Let Z be a Banach space with basis (z i ) such that (x i, f i ) is strongly shrinking relative to (z i ). Then for all k N and ε > 0, there exists N N such that sup k n m n ( x (x j )f j (x i ))zi < ε x i=n for all x X Proof. Let k N and ε > 0. By renorming Z, we may assume without loss of generality that (z i ) is bimonotone. Let K 1 be the frame constant of the frame (f i, x i ) for X. We ε choose a finite -net 2 S (y α) α A in {y K B Y : y span 1 i k (f i )}. By Theorem 2.5, the bounded operator S : X [zi ] is given by S (x ) = x (x i )zi for all x X. As (zi ) is a basis for [z ], for each α A, there exists N α N such that P [Nα, ) S (yα) = i=n α yα(x i )zi < ε. We set N = max 2 α A N α. Given, x BX and m, n N such that k n m, we choose α A such that yα n x (x j )f j < ε, which yields the following 2 S estimates. n ( x (x j )f j (x i ))zi n = P [N, ) S ( x (x j )f j ) i=n P [N, ) S (y α) + P [N, ) S (y α P [N, ) S (yα) + P [N, ) S yα < ε 2 + S ε 2 S = ε n x (x j )f j ) n x (x j )f j Lemma Let X be a Banach space with a shrinking Schauder frame (x i, f i ) and let Z be a Banach space with a basis (z i ). Then for all k N and ε > 0, there exists N N such that sup n m N k n ( x (x j )f j (x i ))zi < ε x for all x X Proof. Let k N and ε > 0. As (x j, f j ) j=1 is a Schauder frame for X, the series j=1 f j(x i )x j converges in norm to x i for all i N. Thus there exists N N such that sup n m N n f j(x i )x j < 11

12 ε k z i for all 1 i k. For x B X k n ( x (x j )f j (x i ))zi and n m N, we have that k n z f j (x i )x j i < k ε k z i z i = ε The following lemma and theorem are based on an idea of W. B. Johnson [J], and are analogous to Proposition 3.1 in [FOSZ], and Lemma 4.3 in [OS]. Their importance comes from allowing us to use arguments that require skipping coordinates, and in particular, will allow us to apply Proposition Lemma Let X be a Banach space with a boundedly complete Schauder frame (x i, f i ) X X. Let ε i 0 and (p i ) [N] ω. There exists (k i ) [N] ω such that for all x X and for all N N there exists M N such that k N < M < k N+1 and sup p M+1 >n m p M 1 n x (f i )x i < εn x. i=m Proof. Assume not, then there exists ε > 0 and K 0 N such that for all K > K 0 there exists x K B X such that for all K 0 < M < K there exists n K,M, m K,M N with p M 1 m K,M n K,M < p M+1 and n K,M i=m K,M x K (f i)x i > ε. As [p M 1, p M+1 ] is finite, we may choose a sequence (K i ) [N] ω such that for every M N there exists n M, m M N such that n Ki,M = n M and m Ki,M = m M for all i M. After passing to a further subsequence of (K i ), we may assume that there exists x X such that x K i (f j ) x (f j ) for all j N. Thus n M i=mm x (f i )x i ε. This contradicts that the series x (f i )x i is norm convergent. Theorem Let X be a Banach space with a shrinking Schauder frame (x i, f i ). Let Z be a Banach space with basis (z i ) such that (x i, f i ) is strongly shrinking relative to (z i ). Let (p i ) [N] ω and (δ i ) (0, 1) with δ i 0. Then there exists (q i ), (N i ) [N] ω such that for any (k i ) i=0 [N] ω and y S X, there exists yi X and t i (N ki 1 1, N ki 1 ) for all i N with N 0 = 0 and t 0 = 0 so that the following hold (a) y = y i and for all l N we have (b) either yl < δ l or sup pqtl 1 n m n y l (x j)f j < δ l yl and sup n m pqtl n y l (x j)f j < δ l yl, (c) P [pqnkl,p qnkl+1 ) S (yl 1 + y l + y l+1 y ) Z < δ l, where P I is the projection operator P I : [z i ] [z i ] given by P I ( a i z i ) = i I a iz i for all ai z i [z i ] and all intervals I N. Proof. By Theorems 2.1 and 2.5, (f i, x i ) is a boundedly complete frame for X with associated basis (z i ). After renorming, we may assume without loss of generality that (z i ) is 12

13 bimonotone. We let K be the frame constant of (f i, x i ). Let ε i 0 such that 2ε i+1 < ε i < δ i and (1 + K)ε i < δ 2 i+1 for all i N. By repeatedly applying Lemma 2.8 to the frame (x i, f i ) of X, we may choose (q k ) k=1 [N]ω such that for all k N, (5) sup n m p qk+1 >p qk n 0 m 0 n 0 i=m 0 f i ( n f j (x)x j )x i εk x for all x X. By Lemma 2.11, after possibly passing to a subsequence of (q k ) k=1, we may assume that for all k N, p qk n (6) sup ( x (x j )f j (x i ))z i ε k x for all x X. n m p qk+1 By applying Lemma 2.7 to the frame (x i, f i ) of X, after possibly passing to a subsequence of (q k ) k=1, we may assume that for all k N, (7) sup n m p qk+1 >p qk n 0 m 0 n n 0 f i ( f j (x)x j )x i ε k+1 x for all x X. 0 i=m By Lemma 2.10, after possibly passing to a subsequence of (q k ) k=1, we may assume that for all k N, (8) sup p qk >n m 1 i=p qk+1 ( n x (x j )f j (x i ))zi ε k+1 x for all x X. By Lemma 2.12, there exists (N i ) i=0 [N] ω such that N 0 = 0 and for all x X and for all k N there exists t k N such that N k < t k < N k+1 and sup m pqtk 1 n m<pq t k +1 i=n x (x i )f i < ε k x. Let (k i ) i=0 [N] ω and y S X. For each i N, we choose t i (N ki, N ki+1 ) with t 0 = 1 such that n (9) sup y (x j )f j < εi. p qti +1 >n m pq t i 1 We now set y i = pq t i 1 j=p qti 1 y (x j )f j for all i N. We have that, yi = p qti 1 j=p qti 1 y (x j )f j = 13 y (x j )f j = y. j=1

14 Thus (a) is satisfied. In order to prove (b), we let l N and assume that y l > δ l. Let m, n N such that n m p qtl. To prove property (b), we consider the following inequalities. n yl n (x j )f j = K n p qtl 1 y (x i )f i (x j )f j i=p qtl 1 p qtl 1 i=p qtl 1 p qtl 1 i=p qtl 1 p qtl 1 1 y n (x i )f i (x j )f j + y (x i )f i (x j )f j i=p qtl 1 p qtl 1 1 y n (x i )f i + y (x i )f i (x j )f j i=p qtl 1 < Kε tl + ε tl by (9) and (7) < (1 + K)ε tl y l /δ l < (1 + K)ε l y l /δ l < δ l y l. Thus sup n m pqtl n y l (x j)f j < δ l yl, proving one of the inequalities in (b). We now assume that l > 1, and let m, n N such that p qtl 1 n m. To prove the remaining inequality in (b), we consider the following. n yl (x j )f j = p qtl 1 n y (x i )f i (x j )f j i=p qtl 1 n p qtl 1 i=p qtl 1 +1 n p qtl 1 i=p qtl 1 +1 y (x i )f i (x j )f j + n p qtl i=p qtl 1 pqtl y (x i )f i (x j )f j + K i=p qtl 1 < ε tl Kε l 1 by (5) and (9) y (x i )f i (x j )f j y (x i )f i < (ε tl Kε l 1 ) y l /δ l < (1 + K)ε l 1 y l /δ l < δ l y l. 14

15 Thus sup pqtl 1 n m n y l (x j)f j < δ l yl, and hence all of (b) is satisfied. To prove (c), we now consider the following, P [pqnkl,p qnkl+1 )S (yl 1 + yl + yl+1 y ) = Z = Thus (c) is satisfied. p qnkl+1 1 i=p qnkl p qnkl+1 1 i=p qnkl i=p qnkl p qtl 2 1 j=1 p qtl 2 1 j=1 p qtl 2 1 j=1 p qnkl+1 1 i=p qnkl (yl 1 + yl + yl+1 y )(x i )zi y (x j )f j (x i ) + y (x j )f j (x i ) z i j=p qtl+1 y (x j )f j (x i ) zi + y (x j )f j (x i ) zi + < ε Nkl 1 + ε Nkl +1 < ε l by (8) and (6). p qnkl+1 1 p qnkl+1 i=p qnl y (x j )f j (x i ) z i j=p qtl+1 y (x j )f j (x i ) z i j=p qtl+1 Properties of coordinate systems for Banach spaces such as frames, bases and FDDs can impose certain structure on infinite dimensional subspaces. For our purposes, this structure can be intrinsically characterized in terms of even trees of vectors [OSZ1]. In order to index even trees, we define T even = {(n 1,..., n 2l ) : n 1 < < n 2l are in N and l N}. If X is a Banach space, an indexed family (x α ) α T even X is called an even tree. Sequences of the form (x (n1,...,n 2l 1,k)) k=n 2l 1 +1 are called nodes. This should not be confused with the more standard terminology where a node would refer to an individual member of the tree. Sequences of the form (n 2l 1, x (n1,...,n 2l )) l=1 are called branches. A normalized tree, i.e. one with x α = 1 for all α T even, is called weakly null (or w -null) if every node is a weakly null (or w -null) sequence. Given 1 > ε > 0 and A (N S X ) ω, we let A ε = {(l i, yi ) (N S X ) ω : (k i, x i ) A such that k i l i, x i yi < ε2 i i N}, and we let A ε be the closure of A ε in (N S X ) ω. We consider the following game between players S (subspace chooser) and P (point chooser). The game has an infinite sequence of moves; on the n th move S picks k n N and a cofinite dimensional w -closed subspace Z n of X and P responds by picking an element x n S X such that d(x n, Z n ) < ε2 n. S wins the game if the sequence (k i, x i ) the players generate is an element of A 5ε, otherwise P is declared the winner. This is referred to as the (A, ε)-game and was introduced in [OSZ1]. The following proposition is essentially an extension of Proposition 15

16 2.6 in [FOSZ] from FDDs to frames, and relates properties of w -null even trees and winning strategies of the (A, ε)-game to blockings of a frame. Proposition Let X be an infinite-dimensional Banach space with a shrinking Schauder frame (x i, f i ). Let A (N S X ) ω. The following are equivalent. (1) For all ε > 0 there exists (K i ) [N] ω and δ = (δ i ) (0, 1) with δ i 0 and (p i ) [N] ω such that if (yi ) S X and (r i ) i=0 [N] ω so that sup pri 1 +1 n m 1 n y i (x j )f j < δ i and sup n m pri n y i (x j )f j < δ i for all i N then (K ri 1, yi ) A ε. (2) For all ε > 0, S has a winning strategy for the (A, ε)-game. (3) For all ε > 0 every normalized w -null even tree in X has a branch in A ε. Proof. The equivalences (2) (3) are given in [FOSZ]. We now assume (1) holds, and will prove (3). Let ε > 0 and let (x (n 1,...,n 2l ) ) (n 1,...,n 2l ) T even be a w -null even tree in X. There exists (K i ) [N] ω and δ = (δ i ) (0, 1) with δ i 0 and (p i ) [N] ω such that if (yi ) S X and (r i ) i=0 [N] ω so that sup pri 1 +1 n m 1 n y i (x j )f j < δ i and sup n m pri n y i (x j )f j < δ i for all i N then (K ri 1, yi ) A ε. We shall construct by induction sequences (r i ) i=0, (n i ) [N] ω such that K ri = n 2i+1 and sup pri 1 +1 n m 1 n x (n 1,...,n 2i ) (x j)f j < δ i and sup n m pri n y i (x j )f j < δ i for all i N. To start, we let r 0 = 1 and n 1 = K 1. Now, if l N and (r i ) l i=0 and (n i ) 2l+1 have been chosen, then using that (x (n 1,...,n 2l+1,k) ) k=n 2l+1 +1 is w -null, we may choose n 2l+2 > n 2l+1 such that x (n 1,...,n 2l+1,n 2l+2 ) (x j)f j < (p rl +1) 1 δ l+1. Thus, sup prl +1 n m 1 n x (n 1,...,n 2l+1,n 2l+2 ) (x j)f j < δ l+1. As (x j, f j ) is a Schauder frame, we may choose r l+1 > r l such that sup n m p rl+1 n x (n 1,...,n 2l+1,n 2l+2 )(x j )f j < δl+1. We then let n 2l+2 = K rl+1. Thus, our sequences (r i ) i=0 and (n i ) may be constructed by induction to satisfy the desired properties, giving us that (n 2i 1, x (n 1,...,n 2i ) ) = (K ri 1, x (n 1,...,n 2i ) ) A ε. We now assume (2) holds, and will prove (1). Let ε > 0 and assume that player S has a winning strategy for the (A, ε)-game. That is, there exists an indexed collection (k (x 1,...,x l ) ) (x 1,...,x l ) X<N of natural numbers, and an indexed collection (X(x 1,...,x )) (x of co-finite dimensional l 1,...,x l ) X<N w -closed subsets of X such that if (x i ) S X and d(x i, X(x 1,...,x )) < 1 i 10 ε2 i for all i N then (k (x 1,...,x i ), X(x 1,...,x i )) A ε/2 and (k (x 1,...,x i ) ) [N] ω. We construct by induction (K i ) [N] ω, (p i ) [N] ω, (δ i ) (0, 1) ω and a nested collection (D i ) [X <ω ] ω such that D i is 1 20 ε2 i -dense in [f j ] p i j=1 and if (y i ) S X and (r i ) i=0 [N] ω so that sup pri 1 +1 n m 1 n y i (x j )f j < δ i and sup n m pri n y i (x j )f j < δ i for all i N, and x i D ri+1 such that yi x i < 1 20 ε2 i for all i N, then K ri 1 k (x 1,...,x i 1 ), and d(x i, X(x )) < 1 1,...,x i 10 ε2 i. This would give that (k (x 1,...,x i ), X(x 1,...,x i )) A ε/2. Hence, 16

17 (K ri 1, yi ) A ε as K ri 1 k (x 1,...,x i 1 ) and yi x i < 1 20 ε2 i for all i N. Thus all that remains is to show that (K i ) [N] ω, (p i ) [N] ω, and (D i ) [X <ω ] ω may be constructed inductively with the desired properties. We start by choosing K 1 = k. As (x i, f i ) is a shrinking Schauder frame for X and X X is co-finite dimensional and w -closed, by Lemma 2.8 there exists p 1 N and δ 1 > 0 such that if sup p1 n m 1 n y (x j )f j < δ 1 for some y S X then d(y, X ) < 1 ε. We then let D 20 1 be some finite 1 ε-net in [f 20 i] p 1. Now we assume n N and that (K i ) n [N] <ω, (p i ) n [N] <ω, (δ i ) n (0, 1) <ω and (D i ) n [X <ω ] <ω have been suitably chosen. As (x i, f i ) is a shrinking Schauder frame for X and X(x 1,...,x l ) X is co-finite dimensional and w -closed for all (x 1,..., x l ) [D n] <ω, by Lemma 2.8 there exists p n+1 N and δ n+1 > 0 such that if sup pn+1 n m 1 n y (x j )f j < δ n+1 for some y S X then d(y, (x 1,...,x l ) [Dn]<ωX (x 1,...,x )) < 1 l 20 ε2 n 1. We then let K n+1 = max (x 1,...,x k l ) [Dn]<ω (x 1,...,x l ) and let D n+1 be a finite 1 20 ε2 n 1 -net in [f i ] p n Upper and lower estimates If (x i ) and (v i ) are two sequences in Banach spaces and C R, then we say that (x i ) is C-dominated by (v i ) if a i x i C a i v i for all (a i ) c 00. Let Z be a Banach space with an FDD (E i ), let V = (v i ) be a normalized 1-unconditional basis, and let 1 C <. We say that (E i ) satisfies subsequential C-V -upper block estimates if every normalized block sequence (z i ) of (E i ) in Z is C-dominated by (v mi ), where m i = min supp E (z i ) for all i N. We say that (E i ) satisfies subsequential C-V -lower block estimates if every normalized block sequence (z i ) of (E i ) in Z C-dominates (v mi ), where m i = min supp E (z i ) for all i N. We say that (E i ) satisfies subsequential V -upper (or lower) block estimates if it satisfies subsequential C-V -upper (or lower) block estimates for some 1 C <. Note that if (E i ) satisfies subsequential C-V -upper block estimates and (z i ) is a normalized block sequence with max supp E (z i 1 ) < k i min supp E (z i ) for all i > 1, then (z i ) is C-dominated by (v ki ) (and a similar remark holds for lower estimates). The following well known Lemma shows why upper estimates are useful for proving theorems about shrinking basic sequences and why lower estimates are useful for proving theorems about boundedly complete basic sequences. Lemma 3.1. Let (E i ) be an FDD and let (v i ) be a normalized basic sequence. If (v i ) is weakly null and (E i ) satisfies subsequential V -upper block estimates then (E i ) is shrinking. If (v i ) is boundedly complete and (E i ) satisfies subsequential V -lower block estimates then (E i ) is boundedly complete. Proof. We first assume that (v i ) is weakly null and (E i ) satisfies subsequential C-V -upper block estimates for some C > 0. Let (z i ) be a normalized block sequence of (E i ). We have that (z i ) is C-dominated by (v mi ) where m i = min supp E (z i ) for all i N. Let ε > 0. Because (v i ) is weakly null there exists (λ i ) n (0, ) such that λ i = 1 and 17

18 λ i v mi < ε/c. Hence, λ i z i C λ i v i < ε. We have that 0 is contained in the closed convex hull of every normalized block sequence of (E i ) and hence (E i ) is shrinking. We now assume that (v i ) is boundedly complete and (E i ) satisfies subsequential V - lower block estimates. Let (z i ) be a semi-normalized block sequence of (E i ). We have that (v mi ) is dominated by (z i ) where m i = min supp E (z i ) for all i N. As (v mi ) is boundedly complete, sup n N n v m i =. Hence, sup n N n z i =, and thus (E i ) is boundedly complete as well. Subsequential V ( ) -upper block estimates and subsequential V -lower block estimates are dual properties, as shown in the following proposition from [OSZ1]. Proposition 3.2. [OSZ1, Proposition 2.14] Assume that Z has an FDD (E i ), and let V = (v i ) be a normalized and 1-unconditional basic sequence. The following statements are equivalent: (a) (E i ) satisfies subsequential V -lower block estimates in Z. (b) (E i ) satisfies subsequential V ( ) -upper block estimates in Z ( ). (Here subsequential V ( ) -upper estimates are with respect to (v i ), the sequence of biorthogonal functionals to (v i ) ). Moreover, if (E i ) is bimonotone in Z, then the equivalence holds true if one replaces, for some C 1, V -lower estimates by C-V -lower estimates in (a) and V ( ) -upper estimates by C-V ( ) -upper estimates in (b). Note that by duality, Proposition 3.2 holds if we interchange the words upper and lower. We are interested in creating associated spaces which have coordinate systems satisfying certain upper and lower block estimates. The particular Banach spaces that we start with do not themselves have such coordinate systems, and hence we need to consider intrinsic coordinate free properties of a Banach space which characterizes when a Banach space embeds into a different Banach space with a coordinate system having certain upper and lower block estimates. These intrinsic coordinate free Banach space properties are defined in terms of even trees. Let X be a Banach space, V = (v i ) be a normalized 1-unconditional basis, and 1 C <. We say that X satisfies subsequential C-V -upper tree estimates if every weakly null even tree (x α ) α T even in X has a branch (n 2l 1, x (n1,...,n 2l )) l=1 such that (x (n 1,...,n 2l )) l=1 is C-dominated by (v n 2l 1 ) l=1. We say that X satisfies subsequential V -upper tree estimates if it satisfies subsequential C-V -upper tree estimates for some 1 C <. If X is a subspace of a dual space, we say that X satisfies subsequential C-V -lower w tree estimates if every w -null even tree (x α ) α T even in X has a branch (n 2l 1, x (n1,...,n 2l )) l=1 such that (x (n 1,...,n 2l )) l=1 C-dominates (v n 2l 1 ) l=1. A basic sequence V = (v i ) is called C-right dominant if for all sequences m 1 < m 2 < and n 1 < n 2 < of positive integers with m i n i for all i N the sequence (v mi ) is C-dominated by (v ni ). We say that (v i ) is right dominant if for some C 1 it is C-right dominant. Lemma 3.3. [FOSZ, Lemma 2.7] Let X be a Banach space with separable dual, and let V = (v i ) be a normalized, 1-unconditional, right dominant basic sequence. If X satisfies 18

19 subsequential V -upper tree estimates, then X satisfies subsequential V -lower w tree estimates. Let (x i, f i ) be a shrinking Schauder frame for a Banach space X, and let (v i ) be a normalized, 1-unconditional, block stable, 1-right dominant, and shrinking basic sequence. For any C > 0, we may apply Proposition 2.14 to the set A = {(n i, x i ) (N S X ) ω : (x i ) C dominates (v n i ) } to obtain the following corollary. Corollary 3.4. Let (x i, f i ) be a shrinking Schauder frame for a Banach space X, and let V = (v i ) be a normalized, 1-unconditional, block stable, right dominant, and shrinking basic sequence. The following are equivalent. (1) There exists C > 0, (K i ), (p i ) [N] ω, and δ = (δ i ) (0, 1) with δ i 0 such that if (yi ) S X and (r i ) i=0 [N] ω so that sup m 1 n m pri 1 +1 j=n y i (x j )f j < δ i and sup pri n m m j=n y i (x j )f j < δ i then (yi ) C (vk ri 1 ). (2) X satisfies subsequential V upper tree estimates. Let Z be a Banach space with a basis (z i ), let (p i ) [N] ω, and let V = (v i ) be a normalized 1-unconditional basic sequence. The space Z V (p i ) is defined to be the completion of c 00 with respect to the following norm ZV : a i z ZV M i = max M N,1 r 0 r 1 < <r M p ri+1 1 j=p ri a j z j v ri for (a i ) c 00. Z V The following proposition from [OSZ1] is what makes the space Z V essential for us. Recall that in [OSZ1] a basic sequence, (v i ), is called C-block stable for some C 1 if any two normalized block bases (x i ) and (y i ) with max(supp(x i ), supp(y i )) < min(supp(x i+1 ), supp(y i+1 )) for all i N are C-equivalent. We say that (v i ) is block stable if it is C-block stable for some constant C. We will make use of the fact that the property of block stability dualizes. That is, if (v i ) is a block stable basic sequence then (vi ) is also a block stable basic sequence. Another simple, though important, consequence of a normalized basic sequence (v i ) being block stable, is that there exists a constant c 1 such that (v ni ) is c-equivalent to (v ni+1 ) for all (n i ) [N] ω. Block stability has been considered before in various forms and under different names. In particular, it has been called the blocking principle [CJT] and the shift property [CK] (see [FR] for alternative forms). The following proposition recalls some properties of Z V (p i ) which were shown in [OSZ1]. Proposition 3.5. [OSZ1, Corollary 3.2, Lemmas 3.3, 3.5, and 3.6] Let V = (v i ) be a normalized, 1-unconditional, and C-block stable basic sequence. If Z is a Banach space with a basis (z i ) and (p i ) [N] ω, then (z i ) satisfies 2C-V -lower block estimates in Z V (p i ). If the basis (v i ) is boundedly complete then (z i ) is a boundedly complete basis for Z V (p i ). If the basis (v i ) is shrinking and (z i ) is shrinking in Z, then (z i ) is a shrinking basis for Z V (p i ). 19

20 If U = (u i ) is a normalized, 1-unconditional and block-stable basic sequence such that (v i ) is dominated by (u i ) and (z i ) satisfies subsequential U-upper block estimates in Z, then (z i ) also satisfies subsequential U-upper block estimates in Z V (p i ). Theorem 3.6. Let X be a Banach space with a shrinking Schauder frame (x i, f i ) which is strongly shrinking relative to some Banach space Z with basis (z i ) and bounded operators T : X Z and S : Z X defined by T (x) = f i (x)z i for all x X and S(z) = z i (z)x i for all z Z. Let V = (v i ) be a normalized, 1-unconditional, block stable, right dominant, and shrinking basic sequence. If X satisfies subsequential V upper tree estimates, then there exists (n i ), (K i ) [N] ω such that Z (v K i ) (n i) is an associated space of (f i, x i ) with bounded operators S : X Z(v K ) (n i) and T : Z(v i K ) (n i) X given by S (x ) = x (x i )zi i x X and S (z ) = z (z i )f i for all z Z(v K ) (n i). i for all Proof. After renorming, we may assume that the basis (z i ) is bimonotone. The sequence (f i, x i ) is a boundedly complete Schauder frame for X by Theorem 2.1, and we have that X satisfies subsequential V lower w tree estimates by Lemma 3.3. By Theorem 2.5, the basis (z i ) is an associated basis for (f i, x i ) with bounded operators S : X Z and T : Z X given by S (x ) = x (x i )z i for all x X and S (z ) = z (z i )f i for all z Z. Let ε > 0. By Corollary 3.4, there exists C > 0, (K i ), (p i ) [N] ω, and δ = (δ i ) (0, 1) with δ i 0 and δ i < ε such that if (yi ) S X and (r i ) i=0 [N] ω so that sup 1 n m pri 1 +1 m j=n y i (x j )f j < δ i and sup pri n m m j=n y i (x j )f j < δ i then (y i ) C (v K ri 1 ). We apply Theorem 2.13 to (x i, f i ), (p i ) [N] ω, and (δ i ) (0, 1) to obtain (q i ), (N i ) [N] ω satisfying the conclusion of Theorem By Theorem 2.5, (z i ) is an associated basis for (f i, x i ), and we denote the norm on Z by Z. We block the basis (z i ) into an FDD by setting E i = span j [pqni,p qni+1 )z j for all i N. We now define a new norm Z on span(zi ) by p a i zi M qnri+1 1 = max a j z j vk Z M N,1 r 0 r 1 < <r M Z Nri V j=p qnri for all (a i ) c 00. We let Z be the completion of span(zi ) under the norm Z. Note that z Z z Z for all z Z. As (vi ) is block stable, there exists a constant c 1 such that (vn i ) c (vn i+1 ) for all (n i ) [N] ω. We now show that S (y ) Z (1 + 2ε)3cC y for all y X. Let y X, M N, and 1 r 0 r 1 < < r M. We will show that (1 + 2ε)3cC S y M p qnri+1 1 y (x j )zj vp V Nri By Theorem 2.13, there exists y. i X and t i j=p qnri (N ri 1 1, N ri 1 ) for all i N with N 0 = 0 and t 0 = 0 such that (a) y = y i and for all i N we have 20