A note on compact operators which attain their norm

John M. Baker

Vol. 82 (1979), No. 2, 319–321

DOI: 10.2140/pjm.1979.82.319

Abstract

For Banach spaces X having the unit cell of X^∗∗w^∗-sequentially compact, the compact operators from X into a Banach space Y attain their norm in X^∗∗. The same holds for weakly compact operators if, in addition, X has the strict Dunford-Pettis property. For Banach spaces X such that the quotient space X^∗∗∕X is separable and Y the space of absolutely summable sequences, a proper subset P_σ of the finite rank operators from X into Y is exhibited. The set P_σ is shown to consist of operators which attain their norm and to be norm-dense in the operator space.

Mathematical Subject Classification 2000

Primary: 47B05, 47B05

Secondary: 46B20

Milestones

Received: 4 April 1978

Published: 1 June 1979