Operators with inverses similar to their adjoints
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- by U. N. Singh and Kanta Mangla PDF
- Proc. Amer. Math. Soc. 38 (1973), 258-260 Request permission
Erratum: Proc. Amer. Math. Soc. 45 (1974), 467.
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.References
- W. A. Beck and C. R. Putnam, A note on normal operators and their adjoints, J. London Math. Soc. 31 (1956), 213–216. MR 77905, DOI 10.1112/jlms/s1-31.2.213
- S. K. Berberian, The numerical range of a normal operator, Duke Math. J. 31 (1964), 479–483. MR 164240
- Paul R. Halmos, A Hilbert space problem book, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR 0208368
- James P. Williams, Operators similar to their adjoints, Proc. Amer. Math. Soc. 20 (1969), 121–123. MR 233230, DOI 10.1090/S0002-9939-1969-0233230-5
Additional Information
- © Copyright 1973 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 38 (1973), 258-260
- MSC: Primary 47B15
- DOI: https://doi.org/10.1090/S0002-9939-1973-0310688-5
- MathSciNet review: 0310688