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Reduction of families of operators to integral form. (English. Russian original) Zbl 0643.47036
Sib. Math. J. 28, No. 3, 470-472 (1987); translation from Sib. Mat. Zh. 28, No. 3(163), 149-151 (1987).
Let (X,\(\mu)\) be a space with a complete \(\sigma\)-finite measure \(\mu\), which is also separable (i.e. the space \(L_ 2(X,\mu)\) is separable). The author shows that if for a family \(\{T_{\alpha};\alpha \in A\}\) of linear and continuous operators acting in \(L_ 2(X,\mu)\) there exists an orthonormal sequence \(\{h_ n\}\subset L_ 2(X,\mu)\) such that \(\lim_{n\to \infty}\sup_{a\in A}\| T\) \(*h_ n\| =0\), then there exists a unitary operator U in \(L_ 2(X,\mu)\) such that UZ and \(UZU^{-1}\) are integral operators (satisfying the Carleman condition) for every Z in the linear hull of the family \(\{T_{\alpha}H_{\alpha};\alpha \in A\}\) where \(H_{\alpha}\) are arbitrary linear and continuous operators acting in \(L_ 2(X,\mu)\). Some consequences of this result to solving linear equations are also given.
Reviewer: F.H.Vasilescu
MSC:
47B38 Linear operators on function spaces (general)
47Gxx Integral, integro-differential, and pseudodifferential operators
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References:
[1] P. R. Halmos, Measure Theory, van Nostrand, New York (1950).
[2] V. B. Korotkov, Integral Operators [in Russian], Nauka, Novosibirsk (1983). · Zbl 0526.47015
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