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

Feder, Moshe

Canadian mathematical bulletin, Tome 25 (1982) no. 1, pp. 78-81

Résumé

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 {T i} 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).

Détail

DOI : 10.4153/CMB-1982-011-0

Mots-clés : 46A32, 46B20, 46B25, 47D15, Banach spaces, compact operators, projections, unconditional compact expansion