Classifying special operators by means of subsets associated with the numerical range

Mary Rodriguez Embry

Vol. 38 (1971), No. 1, 61–65

DOI: 10.2140/pjm.1971.38.61

Abstract

Let A be a continuous linear operator on a complex Hilbert space X, with inner product <,> and associated norm ∥∥. For each complex number z let M_z(A) = {x : ⟨Ax,x⟩ = z∥x∥²}. The following classifications of special operators are obtained: (i) A is a scalar multiple of an isometry if and only if AM_z(A) ⊂ M_z(A) for each complex z; (ii) A is a nonzero scalar multiple of a unitary operator if and only if AM_z(A) = M_z(A) for each complex z; and (iii) A is normal if and only if for each complex z{x|Ax ∈ M_z(A)} = {x|A^∗x ∈ M_z(A)}.

Mathematical Subject Classification 2000

Primary: 47B15

Milestones

Received: 16 June 1970

Revised: 5 October 1970

Published: 1 July 1971