Vol. 20, No. 1, 1967

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On operators whose spectrum lies on a circle or a line

Herbert Meyer Kamowitz

Vol. 20 (1967), No. 1, 65–68

The purpose of this note is to prove the following theorem. THEOREM. Let N be a bounded linear operator on Hilbert space H satisfying

∥N T − T N ∥ = ∥N ∗T − T N∗∥

for all bounded linear operators T. Then N is (obviously) normal and the spectrum of N lies on a circle or straight line.

Here N denotes the adjoint of the operator N.

It is clear that if S is a unitary or self-adjoint operator and α and β are complex numbers, then N = αI + βS satisfies (1). The theorem asserts that the converse is also true.

Mathematical Subject Classification
Primary: 47.40
Received: 27 April 1965
Published: 1 January 1967
Herbert Meyer Kamowitz