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

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.