Complemented Subspaces of Linear Bounded Operators

Abstract

Y ). Emmanuele and John showed that if c 0 ֒→ K(X, Y ), then K(X, Y ) is uncomplemented in L(X, Y ). Bator and Lewis showed that if X is not a Grothendieck space and c X, Y ). In this paper, classical results of Kalton and separably determined operator ideals with property ( * ) are used to obtain complementation results that yield these theorems as corollaries.

A note on Dunford-Pettis like properties and complemented spaces of operators

“…Proof: The proof is similar to the proof of [1,Theorem 20], replacing "relatively weakly p-compact" with "relatively compact".…”

“…A closed operator ideal O has property ( * ) if whenever X is a Banach space and A / ∈ O(X, ℓ ∞ ), then there is an infinite subset M 0 of N such that A M / ∈ O(X, ℓ ∞ ) for all infinite subsets M of M 0 , see [1]. Proof: The idea for the proof comes from Theorem 2.17 in [1]. Let A : X → ℓ ∞ be an operator which is not p-convergent.…”

“…see [10], [9], [17], [12], and [11]. In [1] the authors studied the complementability of the space W(X, ℓ ∞ ) in L(X, ℓ ∞ ). It was shown that if X is not reflexive, then W(X, ℓ ∞ ) is not complemented in L(X, ℓ ∞ ), see [1,Theorem 3].…”

“…In [1] the authors studied the complementability of the space W(X, ℓ ∞ ) in L(X, ℓ ∞ ). It was shown that if X is not reflexive, then W(X, ℓ ∞ ) is not complemented in L(X, ℓ ∞ ), see [1,Theorem 3]. Let CC(X, Y ), Lcc(X, Y ), or LC p (X, Y ) denote the set of all completely continuous, limited completely continuous, or limited p-convergent, respectively, operators from X to Y .…”

“…Note that X 0 is a separable subspace of X and S| X0 is not completely continuous. By [1,Theorem 2.17], the ideal of completely continuous operators has property ( * ). Let M 0 be an infinite subset of N so that…”