Operators with powers essentially similar to those of their adjoints

Patel, S. M.

doi:10.1090/S0002-9939-1974-0326453-X

Operators with powers essentially similar to those of their adjoints
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by S. M. Patel PDF

Proc. Amer. Math. Soc. 42 (1974), 243-247 Request permission

Abstract:

Let T be an operator on a Hilbert space H. In the present note the following result is obtained: If T is an operator such that for some integers $p,q,S{T^{ \ast p}} = {T^q}S + K$, where 0 is not in the essential numerical range of S, and K is compact, then for any complex number $\lambda$ in the essential spectrum of T, ${\lambda ^{ \ast p}} = {\lambda ^q}$.

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On powers of an operator similar to those of its adjoints

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