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Proceedings of the American Mathematical Society

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

 

 

Seminormal operators with compact self-commutators


Author: Kevin Clancey
Journal: Proc. Amer. Math. Soc. 26 (1970), 447-454
MSC: Primary 47.40
DOI: https://doi.org/10.1090/S0002-9939-1970-0265976-5
MathSciNet review: 0265976
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Abstract: Putnam's inequality $ \pi \vert\vert{T^ \ast }T - T{T^ \ast }\vert\vert \leqq \operatorname{meas}_2({\text{sp}}(T))$ is verified for seminormal operators $ T$ when $ {T^ \ast }T - T{T^ \ast }$ is compact. The spectrum of the seminormal singular integral operator defined for $ f \in {L^2}(0,1)$ by

$\displaystyle Tf(s) = sf(s) + i\left( {\frac{{\phi (s)}}{{\pi i}}\int_0^1 {\frac{{f(t)\bar \phi (t)}}{{s - t}}dt} } \right),\quad s \in [0,1],$

where $ \phi $ is any fixed essentially bounded measurable function on $ [0,1]$, is computed.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1970-0265976-5
Keywords: Seminormal operator, hyponormal operator, subnormal operator, singular integral
Article copyright: © Copyright 1970 American Mathematical Society