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∗∥](a070x.png) | (1) |
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.
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