Geometry of spaces of compact operators

We introduce the notion of compactly locally reflexive Banach spaces and show that a Banach space X is compactly locally reflexive if and only if $\mathcal{K}(Y,X^{**})\subseteq\mathcal{K}(Y,X)^{**}$ for all reflexive Banach spaces Y. We show that X * has the approximation property if and only if X has the approximation property and is compactly locally reflexive. The weak metric approximation property was recently introduced by Lima and Oja. We study two natural weak compact versions of this property. If X is compactly locally reflexive then these two properties coincide. We also show how these properties are related to the compact approximation property and the compact approximation prope…

Bounded approximation properties via integral and nuclear operators

Published version of an article in the journal:Proceedings of the American Mathematical Society. Also available from the publisher, Open Access

Absolutely summing operators on C[0,1] as a tree space and the bounded approximation property

AbstractLet X be a Banach space. For describing the space P(C[0,1],X) of absolutely summing operators from C[0,1] to X in terms of the space X itself, we construct a tree space ℓ1tree(X) on X. It consists of special trees in X which we call two-trunk trees. We prove that P(C[0,1],X) is isometrically isomorphic to ℓ1tree(X). As an application, we characterize the bounded approximation property (BAP) and the weak BAP in terms of X∗-valued sequence spaces.

On lifting the approximation property from a Banach space to its dual

Absolutely summing operators on C[0,1] as a tree space and the bounded approximation property

Abstract Let X be a Banach space. For describing the space P ( C [ 0 , 1 ] , X ) of absolutely summing operators from C [ 0 , 1 ] to X in terms of the space X itself, we construct a tree space l 1 tree ( X ) on X. It consists of special trees in X which we call two-trunk trees. We prove that P ( C [ 0 , 1 ] , X ) is isometrically isomorphic to l 1 tree ( X ) . As an application, we characterize the bounded approximation property (BAP) and the weak BAP in terms of X ∗ -valued sequence spaces.

Strict u-ideals in Banach spaces

We study strict u-ideals in Banach spaces. A Banach space X is a strict u-ideal in its bidual when the canonical decomposition X = X X ? is unconditional. We characterize Banach spaces which are strict u-ideals in their bidual and show that if X is a strict u-ideal in a Banach space Y then X contains c0. We also show that '1 is not a u-ideal.

Bounded approximation properties via integral and nuclear operators

Published version of an article in the journal:Proceedings of the American Mathematical Society. Also available from the publisher, Open Access Let X be a Banach space and let A be a Banach operator ideal. We say that X has the lambda-bounded approximation property for A (lambda-BAP for A) if for every Banach space Y and every operator T is an element of A(X, Y), there exists a net (S-alpha) of finite rank operators on X such that S-alpha -> I-X uniformly on compact subsets of X and lim(alpha) sup parallel to TS alpha parallel to(A)<=lambda parallel to T parallel to(A). We prove that the (classical) lambda-BAP is precisely the lambda-BAP for the ideal I of integral operators, or equivalentl…