Remarks on Weak Compactness of Operators Defined on Certain Injective Tensor Products

Abstract

Abstract.We show that if A" is a -2oo -space with the Dieudonné property and y is a Banach space not containing l\ , then any operator T: X®e Y -> Z , where Z is a weakly sequentially complete Banach space, is weakly compact. Some other results of the same kind are obtained.Let A be a Jz^-space (see [1] for this notion and some useful results on Jz^-spaces) and Y be a Banach space not containing l\ . We consider the injective tensor product X ®£ Y (see [3]), and we investigate the following problem: when is any operator T : X <g>£ Y -> Z, where Z is a Banach space, weakly compact?In the case of X = C{K) there are some papers devoted to the study of this question (see [2,[6][7][8][9]), but nothing seems to be known in the present setting; we observe that the theorems proved in the paper extend all of the above-quoted results, but their proofs make use of the results in [2,8], so that they may be considered interesting complements to those theorems. Because the proofs of our results are similar, we give the proof of Theorem 2 only and leave the others to the reader. We need the following definition: a Banach space E has the Dieudonné property if any weakly completely continuous (or Dieudonné) operator defined on it is weakly compact [8].We recall that C{K) spaces have the Dieudonné property.

Weak*-extreme points of injective tensor product spaces

“…Proof. Since the space A ⊗ $ E * * can be canonically embedded in (A ⊗ $ E) * * [7] we have, for any natural number n A ⊗ $ E (2n) ⊂ (A ⊗ $ E (2n−2) ) * * ⊂ (A ⊗ $ E) (2n) .…”

Strictly Singular and Cosingular Multiplications

“…In a similar manner one gets that K(E) = E * ˆ ǫ E ≈ Eˆ ǫ E * has the DPP whenever E is a L ∞ -space. We sketch here for completeness how to deduce Emmanuele's remark in [E, p. 475] from [B2, Corollary 7] by modifying some ideas from [E,Theorem 2] and [Ci,Theorem 1]. Let S : Eˆ ǫ F → Z be any weakly compact operator, where Z is a Banach space, so that [E,Lemma 1] for the latter isometry), so that the restriction…”

Addendum to: “Remarks on weak compactness of operators defined on certain injective tensor products” [Proc. Amer. Math. Soc. 116 (1992), no. 2, 473–476; MR1120506 (92m:46109)]

“…The idea (used in [1]) of embedding X ®£Y into X** <g>e Y, when X is an Sfoo space can also be used to get the following results. [3] imply that T** is unconditionally converging, too.…”