Mathematika

Research Article

HB-subspaces and Godun sets of subspaces in Banach spaces

Eve Ojaa1

a1 Department of Mathematics, Tartu University, Vanemuise 46, EE2400 Tartu, Estonia.

Abstract

Let X be a Banach space and Y its closed subspace having property U in X. We use a net (Aα) of continuous linear operators on X such that Aα ≤ 1, Aα (X) Y for all α, and limα g(Aαy) = g(y), y Y, gY* to obtain equivalent conditions for Y to be an HB-subspace, u-ideal or h-ideal of X. Some equivalent renormings of c0 and l2 are shown to provide examples of spaces X for which K(X) has property U in L(X) without being an HB-subspace. Considering a generalization of the Godun set [3], we establish some relations between Godun sets of Banach spaces and related operator spaces. This enables us to prove e.g., that if K(X) is an HB-subspace of L(X), then X is an HB-subspace of X**—the result conjectured to be true by Å. Lima [9].

(Received September 22 1995)

Key Words:

  • 46B20: FUNCTIONAL ANALYSIS; Normed linear spaces and Banach spaces, Banach lattices; Geometry and structure of normed linear spaces.