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

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$Banach J. Math. Anal. 2 (2008), no. 2, 1–8 POSITIVITY OF ...$

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 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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