A Remark about the Scalar-Plus-Compact Problem

Summary

1. Introduction

It is an open problem whether there exists an infinite-dimensional Banach space X such that every bounded linear operator from X to itself is of the form ƛI + K, where).ƛ. is a scalar, I is the identity on X and K is a compact operator. The strongest property of a similar nature that has been obtained is that a space may be hereditarily indecomposable (see [GM] for this definition and several others throughout the paper), which implies [GM] that every operator on it is of the form

ƛI + S, where S is strictly singular, and even [Fl] that every operator from a subspace into the space is a strictly singular perturbation of a multiple of the inclusion map. (These results assume complex scalars but several examples are known where the conclusion holds with real scalars.) In this note, we show that the first hereditarily indecomposable space to be discovered [GMJ, which we shall call X, has a subspace Y such that there is a non-compact strictly singular operator from Y into X. Therefore this operator is not a compact perturbation of a multiple of the inclusion map. Since all we are doing is showing that one particular space does not give an example of a stronger property than that required by the problem, the existence of this note needs some justification, which we shall now provide.

First, if one is trying to solve the problem with an example, then a natural line of attack is to try to construct a hereditarily indecomposable space such that every strictly singular operator is compact.