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

 

Pseudo-integral operators


Author: A. R. Sourour
Journal: Trans. Amer. Math. Soc. 253 (1979), 339-363
MSC: Primary 47G05; Secondary 47B38
MathSciNet review: 536952
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Abstract: Let $ (X,\,\mathcal{a},\,m)$ be a standard finite measure space. A bounded operator T on $ {L^2}(X)$ is called a pseudo-integral operator if $ (Tf)(x)\, = \,\int {f(y)\,\mu (x,\,dy)} $, where, for every x, $ \mu (x,\, \cdot \,)$ is a bounded Borel measure on X. Main results: 1. A bounded operator T on $ {L^2}$ is a pseudo-integral operator with a positive kernel if and only if T maps positive functions to positive functions. 2. On nonatomic measure spaces every operator unitarily equivalent to T is a pseudo-integral operator if and only if T is the sum of a scalar and a Hilbert-Schmidt operator. 3. The class of pseudo-integral operators with absolutely bounded kernels form a selfadjoint (nonclosed) algebra, and the class of integral operators with absolutely bounded kernels is a two-sided ideal. 4. An operator T satisfies $ (Tf)(x)\, = \,\int {f(y)\,\mu (x,\,dy)} $ for $ f\, \in \,{L^\infty }$ if and only if there exists a positive measurable (almost-everywhere finite) function $ \Omega $ such that $ \left\vert {(Tf)(x)} \right\vert\, \leqslant \,{\left\Vert f \right\Vert _\infty }\Omega (x)$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1979-0536952-2
PII: S 0002-9947(1979)0536952-2
Article copyright: © Copyright 1979 American Mathematical Society