Linear operators for which T^∗T and T + T^∗ commute

Stephen LaVern Campbell

Vol. 61 (1975), No. 1, 53–57

DOI: 10.2140/pjm.1975.61.53

Abstract

This paper is about the bounded linear operators T acting in a separable Hilbert space h such that T^∗T and T + T^∗ commute. It will be shown that such operators are normal if they are either compact or quasinilpotent. It is conjectured that if T^∗T and T + T^∗ commute, then T = A + Q where A = A^∗, AQ = QA, and Q is quasinormal. This conjecture is shown to be equivalent to [T^∗T − TT^∗]T[T^∗T − TT^∗] being hermitian.

Mathematical Subject Classification 2000

Primary: 47B99

Milestones

Received: 28 May 1975

Revised: 28 July 1975

Published: 1 November 1975