Conditions implying normality in Hilbert space

Mary Rodriguez Embry

Vol. 18 (1966), No. 3, 457–460

DOI: 10.2140/pjm.1966.18.457

Abstract

The problem with which this paper is concerned is that of finding new conditions which imply the normality of an operator on a complete inner product space S. Each such condition, presented in this paper, involves the commutativity of certain operators, associated with a given operator A. Theorem 1 states the equivalence of the following conditions: (i) A is normal, (ii) each of AA^∗ and A^∗A commutes with Re A, (iii) AA^∗ commutes with Re A and A^∗A commutes with Im A. Theorem 2 states that A is normal if AA^∗ and A^∗A commute and Re A is nonnegative definite. Finally, Theorem 3 states that if AA^∗ commutes with each of A^∗A and Re A, then AA^∗ commutes with A. In this case, if A is reversible, then A is normal.

Mathematical Subject Classification

Primary: 47.40

Milestones

Received: 4 June 1965

Published: 1 September 1966