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On the Non-Existence of a Projection onto the Space of Compact Operators

Published online by Cambridge University Press:  20 November 2018

Moshe Feder
Moshe Feder*
Affiliation:
Department of Mathematics University of Toronto, Toronto, Canada M5S 1A1
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Abstract

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Let X and Y be Banach spaces, L(X, Y) the space of bounded linear operators from X to Y and C(X, Y) its subspace of the compact operators. A sequence {Ti} in C(X, Y) is said to be an unconditional compact expansion of T ∈ L (X, Y) if ∑ Tix converges unconditionally to Tx for every x ∈ X. We prove: (1) If there exists a non-compact T ∈ L(X, Y) admitting an unconditional compact expansion then C(X, Y) is not complemented in L(X, Y), and (2) Let X and Y be classical Banach spaces (i.e. spaces whose duals are some LP(μ) spaces) then either L(X, Y) = C(X, Y) or C(X, Y) is not complemented in L(X, Y).

Keywords

46A3246B2046B2547D15Banach spacescompact operatorsprojectionsunconditional compact expansion
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 25 , Issue 1 , 01 March 1982 , pp. 78 - 81
Copyright
Copyright © Canadian Mathematical Society 1982

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