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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Hyponormal operators having real parts with simple spectra


Author: C. R. Putnam
Journal: Trans. Amer. Math. Soc. 172 (1972), 447-464
MSC: Primary 47B20
MathSciNet review: 0310689
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Abstract: Let $ {T^ \ast }T - T{T^ \ast } = D \geq 0$ and suppose that the real part of $ T$ has a simple spectrum. Then $ D$ is of trace class and $ \pi $ trace$ (D)$ is a lower bound for the measure of the spectrum of $ T$. This latter set is specified in terms of the real and imaginary parts of $ T$. In addition, the spectra are determined of self-adjoint singular integral operators on $ {L^2}(E)$ of the form $ A(x)f(x) + \Sigma {b_j}(x)H[f{\bar b_j}](x)$, where $ E \ne ( - \infty ,\infty ),A(x)$ is real and bounded, $ \Sigma \vert{b_j}(x){\vert^2}$ is positive and bounded, and $ H$ denotes the Hilbert transform.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1972-0310689-0
PII: S 0002-9947(1972)0310689-0
Keywords: Hyponormal operators, measure of the spectrum of a hyponormal operator, simple spectra, absolute continuity of operators, trace class operators, spectra of singular integral operators
Article copyright: © Copyright 1972 American Mathematical Society