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

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