Composite integral operators and nuclearity

Abstract

Here the s,(T) are the characteristic numbers ([9J, [16]; so-called s-numbers [12]) of T, defined to be the eigenvalues of the related compact nonnegative selfadjoint operator (T* T) ~/2, arranged in decreasing order and repeated according to multiplicity, Some compact operators between Hilbert spaces have characteristic numbers which are 7-summable for exponents 7 smaller than 2. In the case 7 = 1 the operator is said to be nuclear or of trace class. Since the pioneering work of Grothendieck [13] on nuclear spaces (see also Gohberg and Krein [l 1], [12]; Gel'fand and Vilenkin [10]) it has been of interest to enquire under what various sorts of conditions compact operators are nuclear. For integral operators the following are amongst the known results (see [1], [2], [4], [5], [6], [9], [12], [21], [22], for example):