Vol. 61, No. 2, 1975

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Properties which normal operators share with normal derivations and related operators

Joel Hilary Anderson and Ciprian Foias

Vol. 61 (1975), No. 2, 313–325

Let S and T be (bounded) scalar operators on a Banach space 𝒳 and let C(T,S) be the map on (𝒳), the bounded linear operators on 𝒳, defined by

C (T,S)(X ) = T X − XS

for X in (𝒳). This paper was motivated by the question: to what extent does C(T,S) behave like a normal operator on Hilbert space? It will be shown that C(T,S) does share many of the special properties enioyed by normal operators. For example it will be shown that the range of C(T,S) meets its null space at a positive angle and that C(T,S) is Hermitian if T and S are Hermitian. However, if is a Hilbert space then C(T,S) is a spectral operator if and only if the spectrum of T and the spectrum of S are both finite.

Mathematical Subject Classification 2000
Primary: 47B47
Received: 25 October 1974
Published: 1 December 1975
Joel Hilary Anderson
Ciprian Foias