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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Completely seminormal operators with boundary eigenvalues


Author: Kevin Clancey
Journal: Trans. Amer. Math. Soc. 182 (1973), 133-143
MSC: Primary 47B37
DOI: https://doi.org/10.1090/S0002-9947-1973-0341167-1
MathSciNet review: 0341167
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Abstract: For $ f \in {L^2}(E)$ we consider the singular integral operator $ {T_E}f(s) = sf(s) + {\pi ^{ - 1}}{\smallint _E}f(t){(t - s)^{ - 1}}dt$. These singular integral operators are a special case of operators acting on a Hilbert space with one dimensional self-commutator. We discover generalized eigenfunctions of the equation $ {T_E}f = 0$ and, for $ p < 2$, we will give an $ {L^p}(E)$ solution of the equation $ {T_E}f = {\chi _E}$. The main result of the paper is an example of a nonzero $ {L^2}(E)$ solution of $ {T_E}f = 0$, with $ \lambda = 0$ a boundary point of the spectrum of $ {T_E}$.


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DOI: https://doi.org/10.1090/S0002-9947-1973-0341167-1
Keywords: Singular integral operator, seminormal operator
Article copyright: © Copyright 1973 American Mathematical Society