Table of Contents
Journal of Operators
Volume 2014, Article ID 629502, 6 pages
http://dx.doi.org/10.1155/2014/629502
Research Article

Lattice Trace Operators

School of Mathematics, The University of New South Wales, NSW 2052, Australia

Received 31 October 2013; Accepted 19 March 2014; Published 14 April 2014

Academic Editor: Antun Milas

Copyright © 2014 Brian Jefferies. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A bounded linear operator on a Hilbert space is trace class if its singular values are summable. The trace class operators on form an operator ideal and in the case that is finite-dimensional, the trace tr of is given by for any matrix representation of . In applications of trace class operators to scattering theory and representation theory, the subject is complicated by the fact that if is an integral kernel of the operator on the Hilbert space with a -finite measure, then may not be defined, because the diagonal may be a set of -measure zero. The present note describes a class of linear operators acting on a Banach function space which forms a lattice ideal of operators on , rather than an operator ideal, but coincides with the collection of hermitian positive trace class operators in the case of .