Operators satisfying a G₁ condition

Calvin R. Putnam

Vol. 84 (1979), No. 2, 413–426

DOI: 10.2140/pjm.1979.84.413

Abstract

An operator T on a Hilbert space is said to be G₁ if ∥(T − z)⁻¹∥ = 1∕dist(z,σ(T)) for z∉σ(T) and completely G₁ if, in addition, T has no normal part. Certain results are obtained concerning the spectra of completely G₁ operators and of their real parts. It is shown in particular that there exist completely G₁ operators having spectra of zero Hausdorff dimension. Some sparseness conditions on the spectrum are given which assure that a G₁ operator has a normal part.

Mathematical Subject Classification 2000

Primary: 47B99

Secondary: 47A10

Milestones

Received: 23 September 1977

Published: 1 October 1979