Norm attaining operators on some classical Banach spaces

Walter Schachermayer

Vol. 105 (1983), No. 2, 427–438

DOI: 10.2140/pjm.1983.105.427

Abstract

We construct an operator from L¹[0,1] to C[0,1] which may not be approximated by norm attaining operators with respect to the operator norm. This solves a question raised by J. Johnson and J. Wolfe and furnishes the first example of a pair of classical Banach spaces such that the norm attaining operators are not dense. C[0,1] is the first example of a classical Banach space which does not have property B.

On the other hand, we show that a weakly compact operator from C(K) into a Banach space X may be approximated in norm by norm attaining operators. This shows in particular that the norm attaining operators are dense in B(C(K),L¹[0,1]) and B(C(K),l²), thus solving two questions raised by Johnson and Wolfe.

Mathematical Subject Classification 2000

Primary: 46B25

Secondary: 47B99

Milestones

Received: 1 July 1981

Published: 1 April 1983