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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Operators with inverses similar to their adjoints


Authors: U. N. Singh and Kanta Mangla
Journal: Proc. Amer. Math. Soc. 38 (1973), 258-260
MSC: Primary 47B15
DOI: https://doi.org/10.1090/S0002-9939-1973-0310688-5
Erratum: Proc. Amer. Math. Soc. 45 (1974), 467.
MathSciNet review: 0310688
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Abstract: If $ T$ is an invertible operator on a Hilbert space such that $ {S^{ - 1}}{T^{ - 1}}S = {T^ \ast }$ and $ 0 \notin {\text{Cl}}(W(S))$ for some invertible operator $ S$, where $ {\text{Cl}}(W(S))$ denotes the closure of the numerical range of $ S$ and $ {T^ \ast }$ is the adjoint of $ T$, then it is shown that $ T$ is similar to a unitary operator. In fact, this has been proved as a corollary to a more general result, which also includes the corresponding result of J. P. Williams for selfadjoint operators.


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DOI: https://doi.org/10.1090/S0002-9939-1973-0310688-5
Keywords: Hilbert space, selfadjoint operators, unitary operators, normaloid operators, numerical range
Article copyright: © Copyright 1973 American Mathematical Society