Vol. 82, No. 2, 1979

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A note on compact operators which attain their norm

John M. Baker

Vol. 82 (1979), No. 2, 319–321

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 Pσ of the finite rank operators from X into Y is exhibited. The set Pσ 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
Received: 4 April 1978
Published: 1 June 1979
John M. Baker