Convexity properties of a generalized numerical range

John Emanuel de Pillis

Vol. 39 (1971), No. 3, 767–781

DOI: 10.2140/pjm.1971.39.767

Abstract

A numerical range W_n(A) of a bounded linear operator A on Hilbert space ℋ^p is defined to be the set of complex numbers W_n(A) = {tr(AM) : dimension M = n} where M runs over all orthogonal n-dimensional projections on ℋ, and tr(⋅) is the trace functional. It is known that W_n(A) is always convex (the Hausdorff-Toeplitz theorem tells us that W₁(A) is convex). In what follows, we replace the trace functional by the more general elementary symmetric functions, and derive certain convexity results.

Mathematical Subject Classification 2000

Primary: 47A10

Milestones

Received: 10 October 1969

Revised: 23 September 1970

Published: 1 December 1971