Traces of operators and their history

Abstract

As well known, the trace of an n×n-matrix is dened to be the sum of all entries of the main diagonal. Extending this concept to the infinite-dimensional setting does not always work, since non-converging infinite series may occur. So one had to identify those operators that possess something like a trace. In a first step, integral operators generated from continuous kernels were treated. Then the case of operators on the infinite-dimensional separable Hilbert space followed. The situation in Banach spaces turned out to be more complicated, since the missing approximation property causes a lot of trouble. To overcome those difficulties, we present an axiomatic approach in which operator ideals play a dominant rule. The considerations include also singular traces that, by denition, vanish on all finite rank operators.