A Brief Survey on

Abstract

This paper surveys the study of diagonals of operators, particularly strictly positive compact operators, in various kinds of orbits like unitary orbits, and its interplay with the development of infinite majorization theory, stochastic matrices and arithmetic mean ideals of operators. It includes a description of the author’s main results joint with V. Kaftal in [13] and [14] and a brief description of the history including contributions from Hardy–Littlewood–Pólya, Schur and Horn, Gohberg and Markus, A. Neumann, Antezana–Massey–Ruiz–Stojanoff, and Arveson and Kadison. For a more extensive list of historical and related references, see [13] and [14]. This survey on B(H)-subideals [22], which paper is joint with S. Patnaik, is a brief description of the 1983 Fong–Radjavi characterization of those principal ideals in K(H) that are also B(H) ideals and our recent generalization of this to arbitrary B(H)-ideals along with characterization of their structure. Central to this is a notion of soft ideals.

Mathematics Subject Classification (2010). Primary: 15A51, 47A75, 47L20, 47B10, 47B07; Secondary: 47B47, 47B37, 13C05, 13C12.