HB-subspaces and Godun sets of subspaces in Banach spaces

Let X be a Banach space and Y its closed subspace having property U in X. We use a net (A,) of continuous linear operators on X such that \\A(a)\\less than or equal to 1, A(a)(X)subset of Y for all a, and lim(a) g(A(a)y)=g(y), y is an element of Y, g is an element of Y*, to obtain equivalent conditions for Y to be an HB-subspace, u-ideal or h-ideal of X. Some equivalent renormings of c(0) and l(2) 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 Angstrom. Lima [9].