Frames and Riesz bases for Banach spaces, and Banach spaces of ...

Banach J. Math. Anal. 7 (2013), no. 2, 172–193Banach Journal of Mathematical AnalysisISSN: 1735-8787 (electronic)www.emis.de/journals/BJMA/FRAMES AND RIESZ BASES FOR BANACH SPACES, ANDBANACH SPACES OF VECTOR-VALUED SEQUENCESKYUGEUN CHO 1 , JU MYUNG KIM 2∗ , AND HAN JU LEE 3Communicated by I. B. JungAbstract. This paper is devoted to an investigation of frames and Riesz basesfor general Banach sequence spaces. We establish various relationships betweenBessel (respectively, frames) and Riesz sequences (respectively, Riesz bases),and then some of their applications are presented. Some recent results forBanach frames and atomic decompositions are sharpened with simple proofs.Banach spaces consisting of Bessel or Riesz sequences are introduced and itis shown that they are isometrically isomorphic to some Banach spaces ofbounded linear operators, and that some subspaces of those Banach spaces areisometrically isomorphic to some Banach spaces of compact operators.1. IntroductionA sequence (f n ) in a Hilbert space H is called a frame if there exist constantsA, B > 0 such that( ∑ ) 1A‖f‖ ≤ |〈f, f n 〉| 2 2≤ B‖f‖nfor every f ∈ H. The concept is well known and the theory for it has beenmuch studied. The introductory text of Christensen [9] and the survey article ofCasazza [6] contain many results and references for the frame theory for Hilbertspaces. We are now naturally led to a Banach space version of the frame. ForDate: Received: 20 October 2012; Accepted: 23 January 2013.∗ Corresponding author.2010 Mathematics Subject Classification. Primary 46B15; Secondary 46B28, 46B45.Key words and phrases. Banach sequence space, frame, Riesz basis, atomic decomposition,approximation property.172

FRAMES AND RIESZ BASES FOR BANACH SPACES 1731 ≤ p ≤ ∞, a sequence (x ∗ n) in the dual space X ∗ of a Banach space X is calleda p-frame for X if there exist constants A, B > 0 such that( ∑ ) 1A‖x‖ ≤ |x ∗ n(x)| p p≤ B‖x‖nfor every x ∈ X. For the case p = ∞, ( ∑ n |x∗ n(x)| p ) 1 p is replaced by supn |x ∗ n(x)|.If there exists a 2-frame for a Banach space, then the Banach space is isomorphicto a Hilbert space. The concept was introduced by Aldroubi, Sun and Tang [2]and some abstract theories for it were studied by Christensen and Stoeva [11, 19].Casazza, Christensen and Stoeva [7] introduced and studied a more generalnotion. They defined that a sequence (x ∗ n) in X ∗ is a B s -frame for X, where B sis a scalar-valued Banach sequence space that is a linear space of sequences witha norm which makes it a Banach space and for which the coordinate functionalsare continuous, if(i) (x ∗ n(x)) ∈ B s for every x ∈ X,(ii) there exist constants A, B > 0 such thatA‖x‖ ≤ ‖(x ∗ n(x))‖ Bs ≤ B‖x‖for every x ∈ X. An l p -frame for a Banach space is exactly a p-frame. We saythat (x ∗ n) is a B s -Bessel sequence for X if (i) and the upper B s -frame conditionare satisfied. Since a Banach space X can be identified with a subspace of thebidual space X ∗∗ of X, for a given sequence in X, the B s -Bessel sequence (resp.frame) for X ∗ can be analogously defined.For a sequence (x ∗ n) in X ∗ (resp. (x n ) in X), if the mapF (x ∗ n ) : X → B s , x ↦→ (x ∗ n(x))(resp. F (xn) : X ∗ → B s , x ∗ ↦→ (x ∗ (x n )))is well defined, then from the closed graph theorem it is automatically bounded.This means that the condition of B s -Bessel sequence is only (i) in the frameconditions. The operator is called the analysis operator. A sequence is a B s -frame if and only if the analysis operator is an isomorphism.Considering the classical Banach sequence spaces l p (1 ≤ p ≤ ∞) and c 0 , thenan l p (1 ≤ p < ∞) (resp. c 0 )-Bessel sequence (x n ) for X ∗ is a weakly p-summable(resp. null) sequence, an l ∞ -Bessel sequence (x n ) for X ∗ is a bounded sequence,and for a sequence (x ∗ n) in X ∗ , a c 0 -Bessel sequence (x ∗ n) for X is a weak ∗ nullsequence.We say that a sequence (x n ) in X is a B s -Riesz basic sequence for X if(i) ∑ n α nx n converges for every (α n ) ∈ B s ,(ii) there exist constants A, B > 0 such thatA‖(α n )‖ Bs ≤ ∥ ∑ ∥ ∥∥α n x n ≤ B‖(αn )‖ Bsnfor every (α n ) ∈ B s .In particular, a B s -Riesz basic sequence (x n ) for X is a B s -Riesz basis if X =span{x n } and in this paper we call (x n ) a B s -Riesz sequence for X if (i) is satisfied.

174 K. CHO, J.M. KIM, H.J. LEENote that (x n ) is an l ∞ -Riesz sequence for X if and only if (x n ) is unconditionallysummable (cf. [17, Theorem 4.2.8]).For a B s -Riesz sequence (x n ) for X, the synthesis operator is defined byR (xn) : B s → X, (α n ) ↦→ ∑ nα n x n .From the Banach-Steinhaus theorem, a sequence is a B s -Riesz sequence if andonly if the synthesis operator is well defined and bounded. Also a sequence is aB s -Riesz basic sequence if and only if the synthesis operator is an isomorphism.The Riesz basis for a Hilbert space is well known (cf. [6, 9]), and in [2, 11], l p -Rieszbases for Banach spaces were introduced and studied. This paper is organized asfollows.In Section 2 we establish some relationships between Bessel and Riesz sequences.It is well known that a sequence (x n ) in X is an l 1 -Bessel sequencefor X ∗ if and only if (x n ) is a c 0 -Riesz sequence for X (cf. [17, Proposition4.3.9]). Christensen and Stoeva [11, Proposition 2.2] showed that a sequence (x ∗ n)in X ∗ is an l p (1 < p < ∞)-Bessel sequence for X if and only if (x ∗ n) is an l p ∗-Rieszsequence for X ∗ , where p ∗ = p/(p − 1). We extend those results, more precisely,for the dual Banach sequence space Y s of B s , it is shown that a sequence (x n ) inX (resp. (x ∗ n) in X ∗ ) is a Y s -Bessel sequence for X ∗ (resp. X) if and only if (x n )(resp. (x ∗ n)) is a B s -Riesz sequence for X (resp. X ∗ ). Moreover, we establishsome relationships between B s -Bessel and Y s -Riesz sequences.In Section 3 we study some relationships between B s (resp. Y s )-Riesz bases andY s (resp. B s )-frames, necessary and sufficient conditions for Y s (resp. B s )-framesto be B s (resp. Y s )-Riesz bases.In Section 4 we study Banach frames and atomic decompositions. Some recentresults [3, 4, 7] for them are sharpened with simple proofs.We denote the collection of B s -Bessel sequences in X for X ∗ (resp. X ∗ forX) by Bs w (X) (resp. Bs w∗ (X ∗ )). If B s = l p (1 ≤ p < ∞) (resp. B s = l ∞ ),then Bs w (X) is the collection of weakly p-summable (resp. bounded) sequencesin X, and c w 0 (X) (resp. c w∗0 (X ∗ )) is the collection of weakly (resp. weak ∗ ) nullsequences in X (resp. X ∗ ). We denote the collection of B s -Riesz sequences inX by B s R(X). These collections are vector spaces under the standard operationof scalar multiplication and addition for sequences. In Section 5 we show thatthese vector spaces are Banach spaces endowed with some norms and that theyare isometrically isomorphic to some Banach spaces of bounded linear operators.In Section 6 we introduce the ˇB s -Bessel and Riesz sequences which are specialBessel and Riesz sequences. We show that the Banach spaces consisting of themare isometrically isomorphic to some Banach spaces of compact operators. Alsoit is shown that a sequence is a ˇY s -Bessel (resp. ˇBs -Bessel) sequence if and onlyif it is a ˇB s -Riesz (resp. ˇYs -Riesz) sequence.2. Relationships between Bessel and Riesz sequencesThe purpose of this section is to establish some relationships between Besseland Riesz sequences. In order to do this, we need the well known representation

FRAMES AND RIESZ BASES FOR BANACH SPACES 175of the dual space B ∗ s of B s ; cf. [7, Lemma 3.1]. Let (e n ) be the sequence of thecanonical unit vectors and suppose that (e n ) is a Schauder basis for B s . LetY s = {(x ∗ s(e n ))|x ∗ s ∈ B ∗ s}and ‖(x ∗ s(e n ))‖ Ys = ‖x ∗ s‖. Then we see that (Y s , ‖ · ‖ Ys ) is a normed space andthe coordinate functionals for Y s are continuous. Consider the map j s : Y s → Bs∗defined by j s [(x ∗ s(e n ))] = x ∗ s. Then j s is a surjective linear isometry and so Y s isa Banach sequence space. For example, if B s = l p (1 < p < ∞) (resp. l 1 ), thenY s = l p ∗ (resp. l ∞ ), and if B s = c 0 , then Y s = l 1 .Now for every x ∗ s ∈ Bs ∗ and (α n ) ∈ B s( ∑ )x ∗ s((α n )) = x ∗ s α n e n = ∑nnα n x ∗ se n = ∑ nα n (j −1s x ∗ s) n ,where (js−1 x ∗ d ) n is the n-th element of js−1 x ∗ d . Let (f n) be the sequence of thecanonical unit vectors in Y s . Fix k ∈ N. Then for every (α n ) ∈ B sj s f k ((α n )) = ∑ nα n (js−1 j s f k ) n = ∑ nα n (f k ) n = α k .This shows that (j s f n ) is the sequence of the coordinate functionals for B s . If(f n ) is a Schauder basis for Y s , then for every x ∗ s ∈ B ∗ s( ∑ )x ∗ s = j s [(x ∗ s(e n ))] = j s x ∗ s(e n )f n = ∑ nnx ∗ s(e n )j s f n .Throughout this paper we use the objects Y s , j s , (e n ), (f n ), the analysis andsynthesis operators in the introduction. Recall that an operator S from Y ∗ to X ∗is weak ∗ to weak ∗ continuous if and only if there exists an operator T from X toY such that S is the adjoint operator T ∗ of T ; cf. see [17, Theorem 3.1.11]. Wenow haveTheorem 2.1. Suppose that (e n ) is a Schauder basis for B s and let (x n ) be asequence in X. Then the following are equivalent.(a) The analysis operator F (xn) : X ∗ → Y s is well defined.(b) The synthesis operator R (xn) : B s → X is well defined.(c) The analysis operator F (xn) : X ∗ → Y s is well defined and the operator j s F (xn) :X ∗ → Bs ∗ is weak ∗ to weak ∗ continuous.Hence (x n ) ∈ Ys w (X) if and only if (x n ) ∈ B s R(X).Proof. (c)=⇒(a) is trivial.

176 K. CHO, J.M. KIM, H.J. LEE(a)=⇒(b) Recall that F (xn) is bounded. Let (α n ) ∈ B s . Then∥l∑ ∥ ∥∥α n x n =n=msupx ∗ ∈B X ∗∣= sup ∣x ∗ ∈B X ∗l∑α n x ∗ (x n ) ∣n=ml∑α n j s [(x ∗ (x k ))](e n ) ∣n=m(∣l∑ )∣= sup ∣j s [(x ∗ ∣∣(x k ))] α n e nx ∗ ∈B X ∗n=m∥≤ sup ‖j s [(x ∗ ∥∥l∑ ∥ ∥∥Bs(x k ))]‖ B ∗ sα n e nx ∗ ∈B X ∗n=m∥= sup ‖(x ∗ ∥∥l∑ ∥ ∥∥Bs(x k ))‖ Ys α n e nx ∗ ∈B X ∗n=m l∑ ∥ ∥∥Bs= ‖F (xn)‖ ∥ α n e n −→ 0 as l, m → ∞.n=mHence ∑ n α nx n converges.(b)=⇒(c) Recall that R (xn) is bounded. Consider the operator js−1 R(x ∗ : n) X∗ →Y s . Then for every x ∗ ∈ X ∗(x ∗ (x n )) = (x ∗ (R (xn)e n )) = ((R ∗ (x n)x ∗ )e n ) = j −1s R ∗ (x n)x ∗ ∈ Y s .Hence F (xn) is well defined, F (xn) = js−1 R(x ∗ and so j n) sF (xn) = R(x ∗ is n) weak∗ toweak ∗ continuous.□For example, for 1 < p < ∞, (x n ) ∈ l p R(X) if and only if (x n ) ∈ l w p ∗(X),(x n ) ∈ l 1 R(X) if and only if (x n ) ∈ l ∞ (X), and (x n ) ∈ c 0 R(X) if and only if(x n ) ∈ l w 1 (X). For every Banach space X, let (x n ) be a bounded sequence in X,which does not weakly converge to 0. Then (x n ) ∈ l 1 R(X) but (x n ) ∉ c w 0 (X).Remark 2.2. For every Banach space X containing c 0 there exists a (x n ) ∈ l w 1 (X)such that (x n ) ∉ l ∞ R(X) because a Banach space X does not contain c 0 if andonly if every weakly summable sequence in X is unconditionally summable; see[14, (I.4.5)] or [17, Theorem 4.3.12].We have the following duality result of Theorem 2.1.Corollary 2.3. Suppose that (e n ) is a Schauder basis for B s and let (x ∗ n) be asequence in X ∗ . Then the following are equivalent.(a) The analysis operator F (x ∗ n ) : X → Y s is well defined.(b) The synthesis operator R (x ∗ n ) : B s → X ∗ is well defined.(c) The analysis operator F (x ∗ n ) : X ∗∗ → Y s is well defined and the operator j s F (x ∗ n ) :X ∗∗ → Bs ∗ is weak ∗ to weak ∗ continuous.Hence (x ∗ n) ∈ Ysw∗ (X ∗ ) if and only if (x ∗ n) ∈ B s R(X ∗ ) if and only if (x ∗ n) ∈Ys w (X ∗ ).

FRAMES AND RIESZ BASES FOR BANACH SPACES 177Proof. (c)=⇒(a) is clear and (b)=⇒(c) follows from Theorem 2.1(b)=⇒(c).(a)=⇒(b) Since for every (α n ) ∈ B sl∑∣∥ α n x ∗ ∣∣l∑∣n∥ = sup α n x ∗ ∣∣l∑n(x) ∣ = sup α n j s [(x ∗ ∣k(x))](e n ) ∣,x∈B X x∈B Xn=mn=mn=mfrom the proof of Theorem 2.1(a)=⇒(b) the conclusion follows.Interchanging the dual Banach sequence space Y s with B s , we have the followingsymmetric version of Theorem 2.1.Theorem 2.4. Suppose that (e n ) is a Schauder basis for B s and (f n ) is a Schauderbasis for Y s and let (x n ) be a sequence in X. Then the following are equivalent.(a) The analysis operator F (xn) : X ∗ → B s is well defined.(b) The synthesis operator R (xn) : Y s → X ∗∗ is well defined and the operatorR (xn)js−1 : Bs ∗ → X ∗∗ is weak ∗ to weak continuous.(c) The analysis operator F (xn) : X ∗ → B s is well defined and weak ∗ to weakcontinuous.Proof. (c)=⇒(a) is trivial.(a)=⇒(b) Consider the operator F ∗ (x n) j s : Y s → X ∗∗ . Then for every n andx ∗ ∈ X ∗ (F ∗ (x n)j s f n )x ∗ = j s f n F (xn)x ∗ = j s f n [(x ∗ (x k ))] = x ∗ (x n )□and so F(x ∗ j n) sf n = x n for every n. Now for every (β n ) ∈ Y s( ∑ )F(x ∗ n)j s ((β n )) = F(x ∗ n)j s β n f n = ∑ β n F(x ∗ n)j s f n = ∑nnnβ n x n .Hence R (xn) = F(x ∗ j n) s is well defined and R (xn)js−1 = F(x ∗ is n) weak∗ to weakcontinuous because R (xn)(Y s ) ⊂ X.(b)=⇒(c) By the assumption there exists an operator T : X ∗ → B s such thatT ∗ = R (xn)js−1 . But for every x ∗ ∈ X ∗T x ∗ = (j s f n T x ∗ ) = ((T ∗ j s f n )x ∗ ) = (R (xn)f n x ∗ ) = (x ∗ (x n )).Hence F (xn) = T is well defined and weak ∗ to weak continuous because F ∗ (x n) (B∗ s) =R (xn)j −1s (B ∗ s) ⊂ X. □From Theorem 2.4, if (x n ) ∈ B w s (X), then (x n ) ∈ Y s R(X), and the conversedoes not hold in general by Remark 2.2 above, but if B s is reflexive, then theconverse is true.Remark 2.5. The assumption that (f n ) is a Schauder basis for Y s is not usedin the proof of Theorem 2.4(b)=⇒(c), but (a)=⇒(b) need the assumption (seeRemark 2.2).From the same argument as in the proof of Theorem 2.4, we have the followingduality result.Corollary 2.6. Suppose that (e n ) is a Schauder basis for B s and (f n ) is aSchauder basis for Y s and let (x ∗ n) be a sequence in X ∗ . Then the followingare equivalent.

178 K. CHO, J.M. KIM, H.J. LEE(a) The analysis operator F (x ∗ n ) : X → B s is well defined.(b) The synthesis operator R (x ∗ n ) : Y s → X ∗ is well defined and the operatorR (x ∗ n )js−1 : Bs ∗ → X ∗ is weak ∗ to weak ∗ continuous.3. B s -Frames and Riesz basesIn this section we use the results in Section 2 to establish some relationshipsbetween frames and Riesz bases, necessary and sufficient conditions for frames tobe Riesz bases.Recall that an operator T is surjective if and only if T ∗ is an isomorphism,and T ∗ is surjective if and only if T is an isomorphism; cf. [17, Theorem 3.1.22].Then we haveTheorem 3.1. Suppose that (e n ) is a Schauder basis for B s and let (x n ) be asequence in X. Then the following are equivalent.(a) The analysis operator F (xn) : X ∗ → Y s is an isomorphism.(b) The synthesis operator R (xn) : B s → X is surjective.(c) The analysis operator F (xn) : X ∗ → Y s is an isomorphism and the operatorj s F (xn) : X ∗ → B ∗ s is weak ∗ to weak ∗ continuous.Proof. (c)=⇒(a) is trivial.(a)=⇒(b) By Theorem 2.1 R (xn) is well defined. Then R ∗ (x n) = j sF (xn) in theproof of Theorem 2.1(b)=⇒(c). Hence by the assumption (a) R (xn) is surjective.(b)=⇒(c) Since R ∗ (x n) = j sF (xn), by the assumption (b) F (xn) is an isomorphism.□From Theorem 3.1, if (x n ) is a B s -Riesz basis for X, then (x n ) is a Y s -framefor X ∗ . For example, if (x n ) is an l p -Riesz basis for X (1 ≤ p < ∞), then (x n ) isan l p ∗-frame for X ∗ , and if (x n ) is a c 0 -Riesz basis for X, then (x n ) is an l 1 -framefor X ∗ . But a Y s -frame for X ∗ does not imply a B s -Riesz basis for X in general.Indeed, consider the sequence (x n ) = (e 1 , 0, e 2 , 0, · · ·, 0, e n , 0, · · ·) in c 0 . Then forevery (α k ) ∈ l 1 ‖(α k )‖ 1 = ‖((α k )x n )‖ 1 . Thus (x n ) is an l 1 -frame for l 1 , but (x n )fails the condition (ii) of a c 0 -Riesz basic sequence for c 0 .In the proof of Theorem 2.4 the synthesis operator R (xn) : Y s → X ∗∗ is theoperator F ∗ (x n) j s, hence we have the following.Theorem 3.2. Suppose that (e n ) is a Schauder basis for B s and (f n ) is a Schauderbasis for Y s and let (x n ) be a sequence in X. Then the following are equivalent.(a) The analysis operator F (xn) : X ∗ → B s is an isomorphism.(b) The synthesis operator R (xn) : Y s → X ∗∗ is surjective, and the operatorR (xn)j −1s: Bs ∗ → X ∗∗ is weak ∗ to weak continuous.(c) The analysis operator F (xn) : X ∗ → B s is weak ∗ to weak continuous and anisomorphism.Remark 3.3. In Theorem 3.2, if (a) holds, then X is reflexive because R (xn)(Y s ) ⊂X in the proof of Theorem 2.4 . Consequently, under the assumption in Theorem3.2, a nonreflexive Banach space X cannot contain a B s -frame for X ∗ .We now consider sequences in dual spaces. The following result extends [11,Theorem 2.4].

FRAMES AND RIESZ BASES FOR BANACH SPACES 179Theorem 3.4. Suppose that (e n ) is a Schauder basis for B s and (f n ) is a Schauderbasis for Y s and let (x ∗ n) be a sequence in X ∗ . Then the following are equivalent.(a) The analysis operator F (x ∗ n ) : X → B s is an isomorphism.(b) The synthesis operator R (x ∗ n ) : Y s → X ∗ is surjective and the operator R (x ∗ n )js −1 :Bs ∗ → X ∗ is weak ∗ to weak ∗ continuous.Proof. From the same argument as in the proof of Theorem 2.4, we see that thesynthesis operator R (x ∗ n ) : Y s → X ∗ is the operator F(x ∗ ∗ )j s, hence the proof isnestablished.□In view of Corollary 2.6 and Theorem 3.4, if (x ∗ n) ∈ B w∗s (X ∗ ) and (x ∗ n) is a Y s -Riesz basis for X ∗ , then (x ∗ n) is a B s -frame for X, and if B s is reflexive and (x ∗ n)is a Y s -Riesz basis for X ∗ , then (x ∗ n) is a B s -frame for X. Consider the sequence(x ∗ n) = (e 1 , 0, e 2 , 0, · · ·, 0, e n , 0, · · ·) in l 1 . Then (x ∗ n) is a c 0 -frame for c 0 , but (x ∗ n)is not even an l 1 -Riesz basic sequence for l 1 .In Theorem 2.1, we have shown that a sequence (x n ) in X is a B s -Riesz sequencefor X if and only if (x n ) is a Y s -Bessel sequence for X ∗ . But even for a Y s - Rieszbasis in a dual space, it may not be a B s -Bessel sequence.Example 3.5. Let x ∗ 1 = e 1 and for every n ≥ 2 let x ∗ n = (1, 0, · · · , 0, 1, 0, · · · ),where the second 1 is the n-th element. Consider the sequence (x ∗ n) in l 1 . Thenfor every (α n ) ∈ l 1∑n ‖α nx ∗ n‖ 1 = ∑ n 2|α n| < ∞ and so ∑ n α nx ∗ n converges inl 1 ,( ‖(α n )‖ 1 ≤∑)∥ ( ∥∥1 ∞∑ )∥ ∥∥1∥ α k , α 2 , · · · , α n , · · · + ∥ − α k , 0, · · ·and(α n ) =(α 1 −∞∑k=2≤k∥ ∑ nα n x ∗ n∥ +1∞∑|α k | ≤ 2∥ ∑ nk=2k=2α n x ∗ n∥ ,1)α k , 0, · · · + (α 2 , α 2 , 0, · · · ) + · · · + (α n , 0, · · · , α n , 0, · · · ) + · · ·=(α 1 −∞∑α k)x ∗ 1 +k=2∞∑α n x ∗ nwhich shows l 1 = span{x ∗ n}. Hence (x ∗ n) is an l 1 -Riesz basis for l 1 . But x ∗ n(e 1 ) = 1for every n and so (x ∗ n) does not weak ∗ converge to 0 in l 1 . Hence (x ∗ n) is not ac 0 -Bessel sequence for c 0 .We next establish necessary and sufficient conditions for Bessel sequences (resp.frames) to be Riesz basic sequences (resp. Riesz bases).Theorem 3.6. Suppose that (e n ) is a Schauder basis for B s and (f n ) is a Schauderbasis for Y s and let (x n ) ∈ Ys w (X). Then the following are equivalent.(a) (x n ) is a B s -Riesz basic sequence for X.(b) For (β n ) ∈ B s , if ∑ n β nx n = 0, then β n = 0 for all n, and R (xn)(B s ) is closedin X.(c) F (xn)(X ∗ ) = Y s .n=2

180 K. CHO, J.M. KIM, H.J. LEE(d) (x n ) has a biorthogonal sequence and F (xn)(X ∗ ) is closed in Y s .(e) For every n x n ∉ span{x i } i≠n and R (xn)(B s ) is closed in X.Proof. (a)=⇒(b) By definition of B s -Riesz basic sequence this is clear.(b)=⇒(a) Since (x n ) ∈ Ys w (X), by Theorem 2.1 the synthesis operator R (xn) :B s → X is bounded. By the assumption (b) R (xn) is injective. Since R (xn)(B s ) isclosed in X, by the open mapping theorem R (xn) is an isomorphism. Hence (a)follows.(a)⇐⇒(c) In the proof of Theorem 2.1 R(x ∗ = j n) sF (xn). Hence the conclusionfollows.(c)=⇒(d) Since F (xn) is surjective, for every n there exists an x ∗ n ∈ F −1(x {f n) n}.Consider the sequence (x ∗ n) in X ∗ and fix k. Thenx ∗ k(x n ) = ϕ n [F (xn)(x ∗ k)] = ϕ n (f k ),where ϕ n is the n-th coordinate functional for Y s . Hence (x ∗ n) is a biorthogonalsequence for (x n ) and so (d) follows.(d)=⇒(c) Let (x ∗ n) be a biorthogonal sequence for (x n ). Then for every nF (xn)x ∗ n = (x ∗ n(x n )) = f n . Consequently {f n } ⊂ F (xn)(X ∗ ). Since (f n ) is aSchauder basis for Y s and F (xn)(X ∗ ) is closed in Y s , Y s = span{f n } = F (xn)(X ∗ ) =F (xn)(X ∗ ).(a)=⇒(e) It is easy to check that if a sequence (x n ) in X is a B s -Riesz basicsequence under the assumption that (e n ) is a Schauder basis for B s , then (x n ) isa Schauder basis for span{x n }. Hence (e) follows.(e)=⇒(b) By the assumption x n ≠ 0 for all n. Suppose that there exists a(β n ) ∈ B s such that ∑ n β nx n = 0 but β m ≠ 0 for some m. Then ∑ n≠m β nx n =−β m x m and so x m = ∑ n≠m − βnβ mx n . Consequently, x m ∈ span{x i } i≠m . Thiscontradicts the assumption (e). Hence (b) follows.□In Theorem 3.6, the condition that (f n ) is a Schauder basis for Y s is only usedin the proof of (d)=⇒(c). From Theorem 3.6 we haveCorollary 3.7. Suppose that (e n ) is a Schauder basis for B s and (f n ) is aSchauder basis for Y s and let (x n ) be a Y s -frame for X ∗ . Then the followingare equivalent.(a) (x n ) is a B s -Riesz basis for X.(b) For (β n ) ∈ B s , if ∑ n β nx n = 0, then β n = 0 for all n.(c) F (xn)(X ∗ ) = Y s .(d) (x n ) has a biorthogonal sequence.(e) For every n x n ∉ span{x i } i≠n .Using the operators in the proof of Theorem 2.4, then from the same argumentas in the proof of Theorem 3.6 we have the following.Theorem 3.8. Suppose that (e n ) is a Schauder basis for B s and (f n ) is a Schauderbasis for Y s and let (x n ) ∈ B w s (X). Then the following are equivalent.(a) (x n ) is a Y s -Riesz basic sequence for X.(b) For (β n ) ∈ Y s , if ∑ n β nx n = 0, then β n = 0 for all n, and R (xn)(Y s ) is closedin X.

FRAMES AND RIESZ BASES FOR BANACH SPACES 181(c) F (xn)(X ∗ ) = B s .(d) (x n ) has a biorthogonal sequence and F (xn)(X ∗ ) is closed in B s .(e) For every n x n ∉ span{x i } i≠n and R (xn)(Y s ) is closed in X.Use the operators in Corollary 2.6 to show Corollary 3.9.Corollary 3.9. Suppose that (e n ) is a Schauder basis for B s and (f n ) is aSchauder basis for Y s and let (x ∗ n) ∈ Bs w∗ (X ∗ ). Then the following are equivalent.(a) (x ∗ n) is a Y s -Riesz basic sequence for X ∗ .(b) F (x ∗ n )(X) = B s .(c) (x ∗ n) has a biorthogonal sequence in X and F (x ∗ n )(X) is closed in B s .Corollaries 3.10 and 3.11 extend [11, Proposition 2.7].Corollary 3.10. Suppose that (e n ) is a Schauder basis for B s and (f n ) is aSchauder basis for Y s and let (x n ) be a B s -frame for X ∗ . Then the following areequivalent.(a) (x n ) is a Y s -Riesz basis for X.(b) For (β n ) ∈ Y s , if ∑ n β nx n = 0, then β n = 0 for all n.(c) F (xn)(X ∗ ) = B s .(d) (x n ) has a biorthogonal sequence.(e) For every n x n ∉ span{x i } i≠n .Corollary 3.11. Suppose that (e n ) is a Schauder basis for B s and (f n ) is aSchauder basis for Y s and let (x ∗ n) be a B s -frame for X. Then the following areequivalent.(a) (x ∗ n) is a Y s -Riesz basis for X ∗ .(b) F (x ∗ n )(X) = B s .(c) (x ∗ n) has a biorthogonal sequence in X.Now we obtain some applications for Riesz bases.Theorem 3.12. Suppose that (e n ) is a Schauder basis for B s and (f n ) is aSchauder basis for Y s . If (x n ) is a B s -Riesz basis for X, then there exists aY s -Riesz basis (x ∗ n) for X ∗ , which is a biorthogonal sequence for (x n ), so thatx = ∑ nfor every x ∈ X and x ∗ ∈ X ∗ .x ∗ n(x)x n ,x ∗ = ∑ nx ∗ (x n )x ∗ nProof. We have shown that if (x n ) is a B s -Riesz basis for X, then (x n ) is aY s -frame for X ∗ and F (xn)(X ∗ ) = Y s . Consider the sequence (F −1(x f n) n) in X ∗ .Then F −1(x f n) n(x k ) = ϕ k (F (xn)F −1(x f n) n) = ϕ k (f n ), where ϕ k is the k-th coordinatefunctional for Y s . Therefore (F −1(x f n) n) is a biorthogonal sequence for (x n ) and forevery x ∗ ∈ X ∗x ∗ = F −1(x F n) (x n)x ∗ = F −1(x n) [(x∗ (x n ))]= F −1(x n)( ∑ )x ∗ (x n )f n = ∑nnx ∗ (x n )F −1(x n) f n.

182 K. CHO, J.M. KIM, H.J. LEELet x ∈ X. Then x = ∑ n β nx n for some sequence (β n ) of scalars and for everyk F −1(x f n) k(x) = ∑ n β nF −1(x f n) k(x n ) = β k . Hence x = ∑ n F −1(x f n) n(x)x n . It isimmediate that (F −1(x f n) n) is equivalent to (f n ). □If B s is reflexive and (x n ) is a Y s -Riesz basis for X, then X is reflexive andso (x n ) is a B s -frame for X ∗ and F (xn)(X ∗ ) = B s by Theorem 3.2 and Corollary3.10. Then by the proof of Theorem 3.12 we haveTheorem 3.13. Suppose that (e n ) is a Schauder basis for B s and (f n ) is aSchauder basis for Y s . Let B s be reflexive. If (x n ) is a Y s -Riesz basis for X,then there exists a B s -Riesz basis (x ∗ n) for X ∗ , which is a biorthogonal sequencefor (x n ), so thatx = ∑ x ∗ n(x)x n , x ∗ = ∑ x ∗ (x n )x ∗ nnnfor every x ∈ X and x ∗ ∈ X ∗ .If (x ∗ n) ∈ Bsw∗ (X ∗ ) (or B s is reflexive) and (x ∗ n) is a Y s -Riesz basis for X ∗ , then(x ∗ n) is a B s -frame for X and F (x ∗ n )(X) = B s by Theorem 3.4 and Corollary 3.11.Then by the proof of Theorem 3.12 we have the following which extends [11,Theorem 2.8].Theorem 3.14. Suppose that (e n ) is a Schauder basis for B s and (f n ) is aSchauder basis for Y s . Let (x ∗ n) ∈ Bsw∗ (X ∗ ) (or B s be reflexive). If (x ∗ n) is aY s -Riesz basis for X ∗ , then there exists a B s -Riesz basis (x n ) for X, which is abiorthogonal sequence for (x ∗ n), so thatx = ∑ nfor every x ∈ X and x ∗ ∈ X ∗ .x ∗ n(x)x n ,x ∗ = ∑ nx ∗ (x n )x ∗ n4. Banach frames and atomic decompositionsFor an operator S : B s → X, a sequence (x n ) in X and a B s -frame (x ∗ n) in X ∗for X, we say that ((x ∗ n), S) (resp. ((x ∗ n), (x n ))) is a Banach frame (BF) (resp.an atomic decomposition (AD)) for X with respect to B s if(S[(x ∗ n(x))] = x resp. ∑ )x ∗ n(x)x n = xnfor every x ∈ X. The BF and AD for Banach spaces were introduced andstudied by Gröchenig [15], and Casazza, Han, Larson, Christensen and Heil [8, 10],respectively. Recently, some abstract theories for them were studied by Carando,Lassalle and Schmidberg [3, 4], and Casazza, Christensen, Stoeva [7]. The purposeof this section is to sharpen some recent results [3, 4, 7] for the BF and AD.First we consider the existence problem of the BF and AD for Banach spaces.In [8, Theorem 2.10], it was shown that there exists an AD for a Banach space Xif and only if X is separable and has the bounded approximation property (BAP).Thus there exist a separable reflexive Banach space Z, which does not have theBAP (see [12]), such that there is no AD for Z. Also we will see that there is noBF for every nonseparable reflexive Banach space from the following proposition.

FRAMES AND RIESZ BASES FOR BANACH SPACES 183Recall that a sequence (x ∗ n) in X ∗ is called total on X if x ∗ n(x) = 0 for every nimplies x = 0.Proposition 4.1. The following are equivalent.(a) There exists a BF for X.(b) X ∗ is weak ∗ separable.(c) X ∗ has a total sequence.Proof. In [7, Lemma 2.6], (c)=⇒(a) was shown and since a weak ∗ countable denseset in X ∗ is a total sequence, (b)=⇒(c) follows.(a)=⇒(b) Let ((x ∗ n), S) be a Banach frame for X with respect to B s . Considerthe countable subset {e ∗ nF (x ∗ n )} ∞ n=1 of X ∗ . If there would exist an x ∗ ∈ X ∗ suchthat x ∗ ∉ span weak∗ {e ∗ nF (x ∗ n )} ∞ n=1, then by the separation theorem there exists anx ∈ X such that span weak∗ {e ∗ nF (x ∗ n )} ∞ n=1 ⊂ ker(x) and x ∗ (x) = 1. Butx = SF (x ∗ n )x = S[(e ∗ nF (x ∗ n )x)] = 0,which is a contradiction. Hence X ∗ = span weak∗ {e ∗ nF (x ∗ n )} ∞ n=1 and so X ∗ is weak ∗separable.□Corollary 4.2. Let X be a reflexive Banach space. Then there exists a BF forX if and only if X is separable.If X is separable, then there exist Banach frames for X and X ∗ . Indeed,if X is separable, then there exist sequences (x n ) in X and (x ∗ n) in X ∗ suchthat (x ∗ n) is total on X and (j X (x n )) is total on X ∗ (see [16, Proposition 1.f.3]),where j X : X → X ∗∗ is the natural isometry. Hence the assertion follows fromProposition 4.1(c)=⇒(a)[7, Lemma 2.6].We now extend some results in [3, 4, 7] for the BF and AD. Our proofs aresimple using some results in the previous sections. First, we note that a B s -Bessel sequence (x ∗ n) in X ∗ for X is a B s -frame if and only if (x ∗ n) is total andF (x ∗ n )(X) is closed in B s . Indeed, if (x ∗ n) is total, then the frame operator F (x ∗ n ) isinjective. Thus if F (x ∗ n )(X) is closed, then by the open mapping theorem F (x ∗ n ) isan isomorphism. We also remark that if there exist operators U : X → B s andV : B s → X such that V Ux = x for every x ∈ X, then ((U ∗ e ∗ n), V ) is a BF forX with respect to B s (see [4]), where each e ∗ n is the coordinate functional for B s .We now establish necessary and sufficient conditions for B s -Bessel sequencesto be Banach frames with respect to B s , which extend [7, Proposition 3.4].Theorem 4.3. Let (x ∗ n) be a B s -Bessel sequence in X ∗ for X. The following areequivalent.(a) F (x ∗ n )(X) is complemented in B s and (x ∗ n) is total.(b) There exists an operator V : B s → X such that V [(x ∗ n(x))] = x for everyx ∈ X.(c) There exists an operator S : B s → X such that ((x ∗ n), S) is a BF for X withrespect to B s .If (e n ) is a Schauder basis for B s , then (a), (b) and (c) are equivalent to(d) There exists a sequence (x n ) in X such that ∑ n α nx n converges for every(α n ) ∈ B s and x = ∑ n x∗ n(x)x n for every x ∈ X.

184 K. CHO, J.M. KIM, H.J. LEE(e) There exists a Y s -frame (x n ) in X for X ∗ such that x = ∑ n x∗ n(x)x n for everyx ∈ X.If (e n ) is a Schauder basis for B s and (f n ) is a Schauder basis for Y s , then theabove statements are equivalent to(f) There exists a Y s -frame (x n ) in X for X ∗ such that x ∗ = ∑ n x∗ (x n )x ∗ n forevery x ∗ ∈ X ∗ .Proof. (a)=⇒(b) Since F (x ∗ n )(X) is closed in B s , by the note above F (x ∗ n ) is anisomorphism. Let P : B s → F (x ∗ n )(X) be a projection. Then F −1(xP is the desired∗ n)operator.(b)=⇒(c) By the assumption V F (x ∗ n )x = x for every x ∈ X. Then by the noteabove ((F(x ∗ ∗ n )e∗ n), V ) is a BF for X with respect to B s . But F(x ∗ ∗ n )e∗ n = x ∗ n for everyn.(c)=⇒(a) By the assumption (x ∗ n) is total and F (x ∗ n )S is a desired projection.(b)=⇒(d) We see that (V e n ) is the desired sequence.(d)=⇒(b) By the assumption we can define the map V : B s → X by V ((α n )) =∑n α nx n . Then by the Banach-Steinhaus theorem V is bounded and then it isthe desired operator.(d)=⇒(e) follows from Theorem 3.1 and (e)=⇒(d) follows from Theorem 2.1.Assume (e). Then (x ∗ (x n )) ∈ Y s for every x ∗ ∈ X ∗ . Hence by Corollary 2.6,(f) follows, and by Theorem 2.1 (f)=⇒(e) follows.□Next, we consider the AD. Remark that an AD for X with respect to a Banachsequence space can be an AD with respect to another Banach sequence space inwhich the sequence of the canonical unit vectors is a Schauder basis for it (see[4]).We say that an AD ((x ∗ n), (x n )) for X is shrinking if for every x ∗ ∈ X ∗∣ ∣∣ ∑sup x ∗ n(x)x ∗ (x n ) ∣ −→ 0x∈B X n≥Nas N → ∞, and that it is boundedly complete if ∑ n x∗∗ (x ∗ n)x n converges for everyx ∗∗ ∈ X ∗∗ . Also it is called unconditional if ∑ n x∗ n(x)x n converges unconditionallyfor every x ∈ X. The concepts were introduced in [3]. We now have thefollowing which extends [4, Remark 3.2].Theorem 4.4. Let ((x ∗ n), (x n )) be an AD for X such that ∑ n |x∗ n(x)x ∗ (x n )| < ∞for every x ∈ X and x ∗ ∈ X ∗ . Then the following are equivalent.(a) X is reflexive.(b) X does not contain an isomorphic copy c 0 and l 1 .(c) X and X ∗ do not contain an isomorphic copy c 0 .(d) ((x n ), (x ∗ n)) is an unconditional AD for X ∗ and ∑ n x∗∗ (x ∗ n)x n unconditionallyconverges for every x ∗∗ ∈ X ∗∗ .(e) ((x ∗ n), (x n )) is shrinking and boundedly complete.Proof. (a)=⇒(b) and (d)=⇒(e) are clear, and (b)=⇒(c) and (e)=⇒(a) followsfrom [16, Proposition 2.e.8] and [3, Proposition 2.4], respectively.(c)=⇒(d) Let x ∗ ∈ X ∗ . Then by the hypothesis and Corollary 2.3 (x ∗ (x n )x ∗ n) ∈l1 w (X ∗ ) and so ∑ n x∗ (x n )x ∗ n is weakly unconditionally Cauchy in X ∗ . By the

FRAMES AND RIESZ BASES FOR BANACH SPACES 185assumption (c) and [17, Theorem 4.3.12] ∑ n x∗ (x n )x ∗ n unconditionally converges.Since for every x ∈ X( ∑ )x ∗ (x n )x ∗ n(x) = x ∗ x ∗ n(x)x n = x ∗ (x),∑n∑n x∗ (x n )x ∗ n = x ∗ unconditionally converges. This shows the first part. Since forevery x ∗ ∈ X ∑ ∑∗ n x∗ (x n )x ∗ n is weakly unconditionally Cauchy, for every x ∗∗ ∈ X ∗∗n x∗∗ (x ∗ n)x n is weakly unconditionally Cauchy in X. By the assumption (c) forevery x ∗∗ ∈ X ∑ ∗∗ n x∗∗ (x ∗ n)x n unconditionally converges.□From the same proof of Theorem 4.4 we have the following which extends [4,Theorem 3.3].Corollary 4.5. Let ((x ∗ n), (x n )) be an AD for X such that ∑ n |x∗ n(x)x ∗ (x n )| < ∞for every x ∈ X and x ∗ ∈ X ∗ . Then the following are equivalent.(a) X ∗ is separable.(b) X does not contain an isomorphic copy l 1 .(c) X ∗ does not contain an isomorphic copy c 0 .(d) ((x n ), (x ∗ n)) is an unconditional AD for X ∗ .(e) ((x ∗ n), (x n )) is shrinking.The following extends [4, Theorem 3.4 and Corollary 3.5].Theorem 4.6. Let ((x ∗ n), (x n )) be an AD for X such that ∑ n |x∗ n(x)x ∗ (x n )| < ∞for every x ∈ X and x ∗ ∈ X ∗ . Then the following are equivalent.(a) X is complemented in X ∗∗ .(b) X does not contain an isomorphic copy c 0 .(c) ∑ n x∗∗ (x ∗ n)x n unconditionally converges for every x ∗∗ ∈ X ∗∗ .(d) ((x ∗ n), (x n )) is boundedly complete.nProof. (c)=⇒(d) is clear and (b)=⇒(c) follows from the proof of Theorem 4.4(c)=⇒(d).(d)=⇒(a) is [4, Remark 2.5] and (a)=⇒(b) is an application of [1, Corollary2.5.9]. □We now apply the AD to the approximation property. We say that X has thethe approximation property (AP) if for every compact subset of X and ε > 0there exists a finite rank operator T on X such that sup x∈K ‖T x − x‖ ≤ ε, and ifwe take the operator T such as ‖T ‖ ≤ λ for some λ ≥ 1, then X is said to havethe bounded approximation property (BAP).An AD ((x ∗ n), (x n )) for X with respect to B s , in which the sequence of thecanonical unit vectors is a Schauder basis for it, is called strongly shrinking [3] iffor every x ∗ ∈ X ∗ {∣ ∣∣ ∑}sup α n x ∗ (x n ) ∣ : ‖(α n )‖ Bs ≤ 1 −→ 0n≥Nas N → ∞. The strongly shrinking property is strictly stronger than shrinking [3,Examples 1.12 and 1.13]. The following is a simple observation of known resultsbut an interesting relation between the AP and AD.Theorem 4.7. The following are equivalent.

186 K. CHO, J.M. KIM, H.J. LEE(a) There exists an AD for X with respect to a Banach sequence space in whichthe sequence of the canonical unit vectors is a shrinking Schauder basis for it.(b) There exists a strongly shrinking AD for X.(c) There exists a shrinking AD for X.(d) X ∗ is separable and has the BAP.(e) X ∗ is separable and has the AP.Proof. (b)=⇒(c) is clear and (a)=⇒(b) follows from [3, Proposition 1.9]. (c)=⇒(d)is [3, Corollary 1.5] and (d)⇐⇒(e) is well known; cf. [5, Theorem 3.6].(d)=⇒(a) From [5, Theorem 4.9] there exists a Banach space Z with a shrinkingbasis (z n ) such that X embeds complementably into Z. Put{B s = (α n )| ∑ }α n z n converges in Znwith ‖(α n )‖ Bs = ‖ ∑ n α nz n ‖ Z . Then we see that B s is a Banach sequence spacein which the sequence of the canonical unit vectors is a shrinking Schauder basisfor it. Since X embeds complementably into Z, we can find an AD for X withrespect to B s .□5. Banach spaces consisting of Bessel or Riesz sequencesRecall the vector spacesB w s (X) = {(x n ) in X : (x ∗ (x n )) ∈ B s for every x ∗ ∈ X ∗ },Bsw∗(X ∗ ) = {(x ∗ n) in X ∗ : (x ∗ n(x)) ∈ B s for every x ∈ X},B s R(X) = {(x n ) in X : ∑ α n x n converges for every (α n ) ∈ B s }.nThen by boundedness of the analysis and synthesis operators, for every (x n ) ∈Bs w (X), (x ∗ n) ∈ Bsw∗ (X ∗ ), (x n ) ∈ B s R(X), respectively,and‖(x n )‖ B w s (X) =are all finite.We now havesupx ∗ ∈B X ∗‖(x ∗ (x n ))‖ Bs , ‖(x ∗ n)‖ B ws∗ (X ∗ ) = sup ‖(x ∗ n(x))‖ Bs ,x∈B X∥ ∥∥ ∑ ∥ ∥∥‖(x n )‖ BsR(X) = sup α n x n(α n)∈B BsProposition 5.1. (Bs w (X), ‖ · ‖ B w s (X)) and (Bsw∗ (X ∗ ), ‖ · ‖ B w ∗s (X )) are Banach∗spaces.Proof. The proofs of the two cases are the same and so we only prove the firstcase. It is easy to check that ‖ · ‖ B w s (X) is a norm on Bs w (X). Let ((x (k)n )) k be aCauchy sequence in Bs w (X) and let m ∈ N be fixed. Let ε > 0 be given. Thenthere exists an N ∈ N so that k, l ≥ N implies‖(x (k)n ) − (x (l)n )‖ B w s (X) ≤ε‖e ∗ m‖ ,n

FRAMES AND RIESZ BASES FOR BANACH SPACES 187where e ∗ m is the m-th coordinate functional for B s . Then k, l ≥ N implies≤‖x (k)m− x (l)m ‖ = sup |x ∗ (x (k)m − x (l)m )|x ∗ ∈B X ∗sup ‖e ∗ m‖‖(x ∗ (x (k)n − x (l)n ))‖ Bs ≤ ε.x ∗ ∈B X ∗Thus (x (k)m ) k is a Cauchy sequence in X and so there exists an x m ∈ X so thatx m (k) −→ x m . We have shown that there exists a sequence (x n ) in X so thatfor every n x (k)n −→ x n as k → ∞. Since every x ∗ ∈ X ∗ ((x ∗ (x n (k) ))) k is aCauchy sequence in B s , for every x ∗ ∈ X ∗ there exists a (αn x∗ ) ∈ B s so that‖(x ∗ (x (k)n )) − (αn x∗ )‖ Bs −→ 0 as k → ∞ and so for every n x ∗ (x (k)n ) −→ αn x∗ .Consequently (x ∗ (x n )) = (αn x∗ ) ∈ B s for every x ∗ ∈ X ∗ and so (x n ) ∈ Bs w (X).Now let ε > 0 be given. Then there exists an N ∈ N so that k, l ≥ N impliessupx ∗ ∈B X ∗‖(x ∗ (x n(k) )) − (x ∗ (x (l)n ))‖ Bs ≤ ε.Then k ≥ N implies that for every x ∗ ∈ B X ∗‖(x ∗ (x (k)n )) − (x ∗ (x n ))‖ Bs = lim ‖(x ∗ (x (k)n )) − (x ∗ (x (l)n ))‖ Bs ≤ ε.lThus k ≥ N implies ‖(x (k)n ) − (x n )‖ B w s (X) ≤ ε. Hence (x (k)n ) −→ (x n ) in Bs w (X)as k → ∞. This completes the proof.□The following theorem shows that they are actually some Banach spaces ofbounded linear operators. Here L is the Banach space of all bounded linearoperators between Banach spaces.Theorem 5.2. For every Banach space X, the following statements hold.(a) Bs w (X) (resp. Bs w∗ (X ∗ )) is isometrically isomorphic to {T ∈ L(X ∗ , B s ) :T ∗ ({e ∗ n} ∞ n=1) ⊂ X} (resp. L(X, B s )).(b) If (e n ) is a Schauder basis for B s , then B s R(X) and Ys w (X) are isometricallyisomorphic to L(B s , X).Proof. (a) Recall the analysis operator F (xn) (resp. F (x ∗ n )) : X ∗ (resp. X) → B sand then we define the map from Bs w (X) (resp. Bs w∗ (X ∗ )) to {T ∈ L(X ∗ , B s ) :T ∗ ({e ∗ n} ∞ n=1) ⊂ X} (resp. L(X, B s )) via(x n ) (resp. (x ∗ n)) ↦→ F (xn) (resp. F (x ∗ n )).Then the maps will be the desired isometries. Let us only check the first case.Since for every n and x ∗ ∈ X ∗ (F(x ∗ n) e∗ n)(x ∗ ) = x ∗ (x n ), F(x ∗ n) ({e∗ n} ∞ n=1) ⊂ X andso the map is well defined and clearly a linear isometry. Now let T be an elementin the codomain space and consider the sequence (T ∗ e ∗ n) in X. Then we see that(T ∗ e ∗ n) ∈ Bs w (X) and F (T ∗ e ∗ n ) = T . Hence the map is surjective.(b) Define the map from B s R(X) to L(B s , X) via (x n ) ↦→ R (xn), where R (xn)is the synthesis operator. Then it is easy to check that the map is a surjectivelinear isometry.We have shown that for a sequence (x n ) in X (x n ) ∈ Ys w (X) if and only if(x n ) ∈ B s R(X) (Theorem 2.1). Moreover, we will show that their norms are the

188 K. CHO, J.M. KIM, H.J. LEEsame, and then conclude that Ys w (X) is isometrically isomorphic to L(B s , X). If(x n ) ∈ B s R(X), then{∥ ∥∥ ∑ ∞ ∥ }∥∥‖(x n )‖ BsR(X) = sup α k x k : (αk ) ∈ B Bsk=1{∣ ∣∣ ∑ ∞ = sup α k x ∗ (x k ) ∣ : (α k ) ∈ B Bs , x ∗ ∈ B X ∗}k=1{∣ ∣∣ ∑ ∞ ∣= sup α k j s [(x ∗ ∣∣(x n ))]e k : (αk ) ∈ B Bs , x ∗ ∈ B X ∗}k=1{∣ ( ∣∣js ∑ ∞ )∣= sup [(x ∗ ∣∣(x n ))] α k e k : (αk ) ∈ B Bs , x ∗ ∈ B X ∗}k=1{∥ }∥∥js= sup [(x ∗ (x n ))] ∥ : x∗ ∈ BBs∗ X ∗= sup{‖(x ∗ (x n ))‖ Ys : x ∗ ∈ B X ∗} = ‖(x n )‖ Y w s (X).For example, l p R(X) and lp w ∗(X) (1 ≤ p < ∞) are isometrically isomorphic toL(l p , X), and c 0 R(X) and l1 w (X) are isometrically isomorphic to L(c 0 , X).Remark 5.3. In Theorem 5.2(a), if (f n ) is a Schauder basis for Y s , then B w s (X) isisometrically isomorphic to the space L w ∗(X ∗ , B s ) of weak ∗ to weak continuousoperators because an operator T : X ∗ → Y is weak ∗ to weak continuous if andonly if T ∗ (Y ∗ ) ⊂ X.Corollary 5.4. Suppose that (e n ) is a Schauder basis for B s and (f n ) is aSchauder basis for Y s . Then Bs w (X ∗ ) is isometrically isomorphic to the spaceW(X, B s ) of weakly compact operators and so Bs w (X ∗ ) is a closed subspace of(X ∗ ) with the same norm.B w∗sProof. Note that for every Banach space X and Y , the space W(X, Y ) is isometricallyisomorphic to the space L w ∗(X ∗∗ , Y ) via T ↦→ j −1YT ∗∗ , where j Y : Y → Y ∗∗is the natural isometry. It follows from Remark 5.3 that Bs w (X ∗ ) is isometricallyisomorphic to the space W(X, B s ). In view of Theorem 5.2(a), the other partfollows.□Recall that for a sequence (x ∗ n) in X ∗ (x ∗ n) ∈ Ysw∗ (X ∗ ) if and only if (x ∗ n) ∈s (X ∗ ) (Corollary 2.3). Moreover, their norms are also the same.Y wCorollary 5.5. Suppose that (e n ) is a Schauder basis for B s . Then Ys w (X ∗ ) andYs w∗ (X ∗ ) are isometrically isomorphic to L(B s , X ∗ ).Proof. By Theorem 5.2(b) we only need to show the case Ysw∗ (X ∗ ). But, for everyBanach space X and Y , L(X, Y ∗ ) is isometrically isomorphic to L(Y, X ∗ ), hencethe conclusion follows from Theorem 5.2(a).□For example, ‖(x ∗ n)‖ l w p (X ∗ )= ‖(x ∗ n)‖ l w ∗p (X ∗ ) (1 ≤ p ≤ ∞) for every (x ∗ n) ∈ l w p (X ∗ ).□

FRAMES AND RIESZ BASES FOR BANACH SPACES 1896. Special Bessel and Riesz sequencesIn this section we consider the following subspaces of Bs w (X), Bsw∗ (X ∗ ), andB s R(X), respectively;ˇB w s (X) = {(x n ) ∈ B w s (X) : limn‖(0, · · ·, 0, x n , x n+1 , · · ·)‖ B w s (X) = 0},ˇB sw∗(X ∗ ) = {(x ∗ n) ∈ Bs w∗(X ∗ ) : lim ‖(0, · · ·, 0, x ∗nn, x ∗ n+1, · · ·)‖ B w ∗s (X ∗ ) = 0},{(x n ) ∈ B s R(X) :ˇB s R(X) ={ ∑}α n x n : (α n ) ∈ B Bsn}is relatively compact in X .We will show that they are actually some Banach spaces of compact operators.In order to do this, we need the following lemma.Lemma 6.1. Suppose that (e n ) is a Schauder basis for B s and let K be a boundedsubset of B s . Then K is relatively compact if and only if lim n sup (kn)∈K ‖(0, · ··, 0, k n , k n+1 , · · ·)‖ Bs = 0.Proof. Let P m : B s → B s be the m-th basis projection for each m ∈ N. Supposethat K is relatively compact. Since for every (α n ) ∈ B s ‖P m (α n ) − (α n )‖ Bs −→ 0as n → ∞ and (P m ) is uniformly bounded, we see that for every ε > 0 there isan N ∈ N such that for all m ≥ Nsup ‖P m (k n ) − (k n )‖ Bs ≤ ε,(k n)∈Khence the assertion follows.Suppose the converse. Let ε > 0. Then there is an N ∈ N such that ‖P N (k n ) −(k n )‖ ≤ ε/2 for all (k n ) ∈ K. Since K is bounded, P N (K) is relatively compactin B s . Let {P N (x 1 ), . . . , P N (x n )} be an ε/2-net of P N (K), where x j ∈ K for all1 ≤ j ≤ n.Then for each x ∈ K, there is a j (1 ≤ j ≤ n) such that ‖P N x − P N x j ‖ ≤ ε/2,and so ‖x − P N x j ‖ ≤ ε. This means that the set {P N (x 1 ), . . . , P N (x n )} is anε-net of K, hence K is relatively compact.□Theorem 6.2. Suppose that (e n ) is a Schauder basis for B s . Then for everyBanach space X, the following statements hold.w∗(a) ˇB s (X ∗ ) is isometrically isomorphic to the space K(X, B s ) of compact operators.(b) ˇB s R(X) is isometrically isomorphic to K(B s , X).(c) If (f n ) is a Schauder basis for Y s , then ˇY s w (X) is isometrically isomorphic toK(B s , X).w∗Proof. (a) Consider the map from ˇB s (X ∗ ) to K(X, B s ) via (x ∗ n) ↦→ F (x ∗ n ). Thenby Lemma 6.1 this map is well defined and clearly a linear isometry. If T ∈K(X, B s ), then by using Lemma 6.1 it is easy to check that (T ∗ e ∗ w∗n) ∈ ˇB s (X ∗ )and F (T ∗ e ∗ n ) = T . Hence the map is surjective.(b) Consider the map from ˇB s R(X) to K(B s , X) via (x n ) ↦→ R (xn). Then it iseasy to check that this map is a surjective linear isometry.

190 K. CHO, J.M. KIM, H.J. LEE(c) By Theorem 6.6(a) ˇB s R(X) = ˇY s w (X) and by the proof of Theorem 5.2(b)their norms are also the same. Hence the conclusion follows from (b). □For example, ľpR(X) and ľw p∗(X) (1 < p < ∞) are isometrically isomorphic toK(l p , X), and č 0 R(X) and ľw 1 (X) are isometrically isomorphic to K(c 0 , X).Proposition 6.3. Suppose that (e n ) is a Schauder basis for B s . Then ˇB w s (X) isa closed subspace of B w s (X).Proof. Let ((x (k)n )) k be a sequence in ˇB s w (X) and let (x n ) ∈ Bs w (X) with ‖(x (k)n ) −(x n )‖ B w s (X) −→ 0 as k → ∞. Consider the analysis operators F , F (k) (x n ) (x n) : X ∗ →B s . Then by Lemma 6.1 each F is a compact operator. Since ‖F −(k) (x n ) (x (k)n )F (xn)‖ = ‖(x (k)n ) − (x n )‖ B w s (X) −→ 0 as k → ∞, F (xn) is a compact operator.Hence by Lemma 6.1 (x n ) ∈ ˇB s w (X).□Now recall the injective tensor product X ˇ⊗Y of Banach spaces X and Y (see[18, Chapter 3] or [13, Section 1.1]). Then X ˇ⊗Y is isometrically isomorphic tothe operator norm closure F w ∗(X ∗ , Y ) of the space of weak ∗ to weak continuousfinite rank operators from X ∗ to Y and, if X has the AP, then F w ∗(X ∗ , Y ) =K w ∗(X ∗ , Y ), the space of weak ∗ to weak continuous compact operators. Then wehaveTheorem 6.4. Suppose that (e n ) is a Schauder basis for B s . Then for everyBanach space X, ˇB w s (X) is isometrically isomorphic to K w ∗(B ∗ s, X).Proof. By the note above, it is enough to show that B s ˇ⊗X is isometrically isomorphicto ˇB w s (X).We define the map J : (B s ⊗ X, ‖ · ‖ ∨ ) → ˇB w s (X) by( ∑ ) (J (λ j i ) ∑ )i ⊗ x j = λ j i x j .j≤nj≤ni

FRAMES AND RIESZ BASES FOR BANACH SPACES 191Then we see that J(B s ⊗ X) ⊂ Bs w (X), and∥(∥∥lim 0, · · · , 0, ∑ λ jmmx j , ∑ )∥λ j ∥∥Bm+1x j , · · ·wj≤n j≤n s (X){∥(∥∥= lim sup 0, · · · , 0, ∑ λ jmmx ∗ (x j ), ∑ )∥ }λ j m+1x ∗ ∥∥Bs(x j ), · · · : x ∗ ∈ B X ∗j≤nj≤n{∣ [( ∣∣γ= lim sup 0, · · · , 0, ∑ λ jmmx ∗ (x j ), ∑ )]∣ }λ j m+1x ∗ ∣∣(x j ), · · · : x ∗ ∈ B X ∗, γ ∈ B B ∗ sj≤nj≤n{∣ ∣∣∑}= lim sup γ[(0, · · · , 0, λ jmmx ∗ (x j ), λ j m+1x ∗ (x j ), · · · )] ∣ : x ∗ ∈ B X ∗, γ ∈ B B ∗ sj≤n∑ {∣ ( ∣∣γ ∑ )∣ }≤ lim sup λ j ∣∣i x∗ (x j )e i : x ∗ ∈ B X ∗, γ ∈ B B ∗msj≤ni≥m∑ {∣ ( ∣∣γ ∑ )∣ }≤ lim ‖x j ‖ sup λ j ∣∣i e i : γ ∈ BB ∗msj≤ni≥m∑∑= lim ‖x j ‖ ∥ λ j i e i∥ = 0.m Bsj≤ni≥mThus J is well defined and linear. Since for every (x n ) ∈ ˇB s w (X) and every m( ∑ )(x 1 , · · · , x m , 0, · · · ) = J e j ⊗ x jand lim m ‖(0, · · · , 0, x m , x m+1 , · · · )‖ B w s (X) = 0, J(B s ⊗ X) is dense in ˇB s w (X).Now∥ ∑ ∥(λ j i ) ∥∥∨i ⊗ x jj≤n{∣ ∣∣ ∑}= γ[(λ j i ) i]x ∗ (x j ) ∣ : x ∗ ∈ B X ∗, γ ∈ B B ∗ sj≤nj≤m{∣ ( ∣∣γ∑ )∣ }= x ∗ (x j )(λ j ∣∣i ) i : x ∗ ∈ B X ∗, γ ∈ B B ∗ sj≤n{∥ ∥∥ ∑}= x ∗ (x j )(λ j i ) i∥ : x ∗ ∈ B X ∗Bsj≤n{∥ ( ∥∥ ∑ ))}=(x ∗ x j λ j i ∥ : x ∗ ∈ B X ∗Bsj≤n( = ∥∑ )x j λ j ij≤n∥i∥i B w s (X).Thus J is an isometry and so there exists an extension ˇJ : B s ˇ⊗X → ˇB s w (X) of Jsuch that ˇJ is surjective and an isometry.□For example, ľw p (X) (1 ≤ p < ∞) (resp. c 0 (X)) is isometrically isomorphic toK w ∗(l p ∗, X) (resp. K w ∗(l 1 , X)).

192 K. CHO, J.M. KIM, H.J. LEECorollary 6.5. Suppose that (e n ) is a Schauder basis for B s . Then for everyBanach space X, the following statements hold.w∗(a) ˇB s (X ∗ ) = ˇB s w (X ∗ ) with the same norm.w∗(b) ˇY s (X ∗ ) = ˇY s w (X ∗ ) with the same norm.Proof. (b) follows from Corollary 5.5, and (a) is a result of Theorems 6.2(a) and6.4 because K(X, B s ) is isometrically isomorphic to K w ∗(Bs, ∗ X ∗ ). □For example, ľw p (X ∗ ) = ľw∗ p (X ∗ ) (1 ≤ p ≤ ∞).Finally we establish relationships between the special Bessel and Riesz sequences.Theorem 6.6. Suppose that (e n ) is a Schauder basis for B s and (f n ) is a Schauderbasis for Y s . Let (x n ) and (x ∗ n) be sequences in X and X ∗ , respectively. Then thefollowing statements hold.(a) (x n ) ∈ ˇY s w (X) if and only if (x n ) ∈ ˇB s R(X).(b) (x ∗ w∗n) ∈ ˇY s (X ∗ ) if and only if (x ∗ n) ∈ ˇB s R(X ∗ ).Proof. (b) follows from (a) and Corollary 6.5(b). To show (a), consider the analysisoperator F (xn) : X ∗ → Y s and synthesis operator R (xn) : B s → X. Then inthe proof of Theorem 2.1 j s F (xn) = R(x ∗ . If (x n) n) ∈ ˇY s w (X), then by Lemma 6.1F (xn) is a compact operator and so is R (xn). Thus (x n ) ∈ ˇB s R(X). Conversely,if (x n ) ∈ ˇB s R(X), then R (xn) is a compact operator and so is F (xn). Hence(x n ) ∈ ˇY s w (X) by Lemma 6.1.□From Theorem 6.6, for a sequence (x n ) in X, (x n ) ∈ ľw p∗(X) if and only if(x n ) ∈ ľpR(X) (1 < p < ∞), and (x n ) ∈ ľw 1 (X) if and only if (x n ) ∈ č 0 R(X).Interchanging B s with Y s we have the following result.Theorem 6.7. Suppose that (e n ) is a Schauder basis for B s and (f n ) is a Schauderbasis for Y s . Let (x n ) and (x ∗ n) be sequences in X and X ∗ , respectively. Then thefollowing statements hold.(a) If (x n ) ∈ Bs w (X), then (x n ) ∈ ˇB s w (X) if and only if (x n ) ∈ ˇY s R(X).(b) If (x ∗ n) ∈ Bsw∗ (X ∗ ), then (x ∗ w∗n) ∈ ˇB s (X ∗ ) if and only if (x ∗ n) ∈ ˇY s R(X ∗ ).Proof. (a) Let (x n ) ∈ Bs w (X). Then by the proof of Theorem 2.4 the analysisoperator F (xn) : X ∗ → B s and synthesis operator R (xn) : Y s → X ∗∗ is well definedand F(x ∗ = R n) (x n)js−1 . Then the conclusion follows from the same argument ofthe proof of Theorem 6.6.(b) Let (x ∗ n) ∈ Bs w∗ (X ∗ ). Then by Corollary 2.6 the analysis operator F (x ∗ n ) :X → B s and synthesis operator R (x ∗ n ) : Y s → X ∗ is well defined, and we see thatF(x ∗ ∗ n ) = R (x ∗ n ) js −1 . Hence the conclusion follows. □Acknowledgement. The authors would like to thank the referee for valuablecomments. The second author was supported by the Korea Research FoundationGrant 2012-0007123 and BK21 Project funded by the Korean Government.Third author was supported by Basic Science Research program through the NationalResearch Foundation of Korea (NRF) funded by the Ministry of Education,Science and Technology (2012R1A1A1006869).

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