Between the unitary and similarity orbits of normal operators

Paul Guinand and Laurent Walsh Marcoux

Vol. 159 (1993), No. 2, 299–335

DOI: 10.2140/pjm.1993.159.299

Abstract

D. A. Herrero has defined the (𝒰 + 𝒦)-orbit of an operator T acting on a Hilbert space ℋ to be (𝒰 + 𝒦)(T) = {R⁻¹TR : R invertible of the form unitary plus compact}. In this paper, we characterize the norm closure in ℬ(ℋ) of such an orbit in three cases: firstly, when T is normal; secondly when T is compact; and thirdly, when T is the unilateral shift. Some consequences of these characterizations are also explored.

Mathematical Subject Classification 2000

Primary: 47A65

Secondary: 47A66, 47B15

Milestones

Received: 1 October 1990

Revised: 20 April 1992

Published: 1 June 1993