Similarities involving normal operators on Hilbert space

Mary Rodriguez Embry

Vol. 35 (1970), No. 2, 331–336

DOI: 10.2140/pjm.1970.35.331

Abstract

The primary purpose of this note is to exhibit a proof and several corollaries of the following theorem concerning continuous linear operators on a complex Hilbert space X.

Theorem 1. If H and K are commuting normal operators and AH = KA, where 0 is not in the numerical range of A, then H = K.

In the entire paper A,E,H and K represent continuous linear operators on X,A^∗ is the adjoint of A,W(A) is the numerical range of A and σ(A) is the spectrum of A. The terms self-adjoint, normal and unitary are used in the standard fashion. A is quasinormal if and only if A commutes with A^∗A. A unitary operator is called cramped if and only if its spectrum is contained in an arc of the unit circle with central angle less than π.

Mathematical Subject Classification 2000

Primary: 47B15

Secondary: 47A99

Milestones

Received: 20 January 1970

Published: 1 November 1970