Integral operators generated by H-continuous measures. (English. Russian original) Zbl 0694.47039
Ukr. Math. J. 41, No. 6, 660-663 (1989); translation from Ukr. Mat. Zh. 41, No. 6, 769-773 (1989).
Let X be a separable Hilbert space and let H be a linear subspace in X. The author introduces the notion of H-compact operator and the notion of H-continuous measure. The main result says that if \(\mu\) is a real-valued measure on X with bounded variation, then \(\mu\) is H-continuous if and only if the operator
\[
(Q_{\mu}f)(x)=\int_{X}f(x-y)d\mu (x)
\]
is H- compact.
Reviewer: V.Petkov
MSC:
| 47Gxx | Integral, integro-differential, and pseudodifferential operators |
| 47B38 | Linear operators on function spaces (general) |
| 46G12 | Measures and integration on abstract linear spaces |
References:
| [1] | A. V. Skorokhod, Integration in Hilbert Space [in Russian], Nauka, Moscow (1975). · Zbl 0355.62084 |
| [2] | A. V. Skorokhod, ?Admissible shifts of measures in Hilbert space,? Teor. Veroyatn. Primen.,15, No. 4, 577-598 (1970). |
| [3] | V. A. Romanov, ?H-continuous measures in a Hilbert space,? Vestn. Mosk. Univ. Mat. Mekh., No. 1, 61-95 (1977). |
| [4] | S. V. Fomin, ?Differentiable measures in linear spaces,? Usp. Mat. Nauk,23, No. 1, 221-222 (1968). · Zbl 0177.18603 |
| [5] | V. A. Romanov, ?Limits of differentiable measures in Hilbert space,? Ukr. Mat. Zh.,33, No. 2, 215-219 (1981). · Zbl 0481.46023 · doi:10.1007/BF01086075 |
| [6] | V. I. Averbukh, O. G. Smolyanov, and S. V. Fomin, ?Generalized functions and differential equations in linear spaces. I. Differentiable measures,? Tr. Mosk. Mat. o-va.,24, 133-174 (1971). · Zbl 0234.28005 |
| [7] | V. A. Romanov, ?Limits of quasiinvariant measures in Hilbert space,? Ukr. Mat. Zh.,31, No. 2, 211-214 (1979). · Zbl 0428.28010 · doi:10.1007/BF01086016 |
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