A decomposition for B(X)^∗ and unique Hahn-Banach extensions

Julien O. Hennefeld

Vol. 46 (1973), No. 1, 197–199

DOI: 10.2140/pjm.1973.46.197

Abstract

For a Banach space X, let B(X) be the space of all bounded linear operators on X, and 𝒞 the space of all compact linear operators on X. In general, the norm-pre-serving extension of a linear functional in the Hahn-Banach theorem is highly non-unique. The principal result of this paper is that, for X = c₀ or l^p with 1 < p < ∞, each bounded linear functional on 𝒞 has a unique norm-preserving to B(X). This is proved by using a decomposition theorem for B(X)^∗, which takes on a special form for X = c₀ or l^p with 1 < p < ∞.

Mathematical Subject Classification

Primary: 47D15

Secondary: 46B05

Milestones

Received: 15 December 1971

Published: 1 May 1973