16 - Selected problems on the structure of complemented subspaces of Banach spaces

Summary

Abstract. We discuss several old and more recent problems on bases, complemented subspaces and approximation properties of Banach spaces

INTRODUCTION AND NOTATION

This paper is based on author's lecture at the conference “Contemporary Ramifications of Banach Space Theory” in honor of Joram Lindenstrauss and Lior Tzafriri, (Jerusalem, June 19-24, 2005). We present several open problems, some more than 20 years old, some recent. They mainly concern properties of various classes of Banach spaces expressed in terms of the structure of complemented subspaces of the spaces, bases and approximation by finite rank operators. The paper consists of 5 sections. In Section 1 we outline the proof of the classical Complemented Subspaces Theorem due to Lindenstrauss and Tzafriri [LT1] which characterizes Hilbert spaces among Banach spaces. We follow the presentation by Kadec and Mityagin [KaMy] emphasizing some points which are omitted in their approach. Section 2 is devoted to some problems which naturally arise from the Complemented Subspace Theorem. In Section 3 we discuss spaces which have few projections; their properties are on the other extreme from the assumption of the Complemented Subspace Theorem. Section 4 is devoted to approximation properties of some special non-separable function spaces and spaces of operators. Section 5 has a differ- ent character; it concerns bases for L^p(G) consisting of characters where G is either a Torus group, or the Dyadic group.

We employ standard Banach space notation and terminology (cf. [LT1]). In particular, “subspace” means “closed linear subspace”, “operator” means “bounded linear operator”, “basis” means “Schauder basis”, “projection’ means “bounded linear idempotent” and “c-complemented subspace” means “the range of projection of norm at most c”.