Pseudo-integral operators
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- by A. R. Sourour PDF
- Trans. Amer. Math. Soc. 253 (1979), 339-363 Request permission
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 $\left | {(Tf)(x)} \right | \leqslant {\left \| f \right \|_\infty }\Omega (x)$.References
- M. B. Abrahamse and Thomas L. Kriete, The spectral multiplicity of a multiplication operator, Indiana Univ. Math. J. 22 (1972/73), 845–857. MR 320797, DOI 10.1512/iumj.1973.22.22072
- William Arveson, Operator algebras and invariant subspaces, Ann. of Math. (2) 100 (1974), 433–532. MR 365167, DOI 10.2307/1970956 —, An invitation to ${C^{\ast }}$-algebras, Springer-Verlag, New York, 1976.
- Robert B. Ash, Real analysis and probability, Probability and Mathematical Statistics, No. 11, Academic Press, New York-London, 1972. MR 0435320
- Frank F. Bonsall and John Duncan, Complete normed algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 80, Springer-Verlag, New York-Heidelberg, 1973. MR 0423029, DOI 10.1007/978-3-642-65669-9 N. Bourbaki, Eléménts de mathématique, Livre VI: Intégration. Chapitre 6: Intégration vectorielle, Hermann, Paris, 1959. N. Dunford and J. T. Schwartz, Linear operators, Part I, Interscience, New York, 1958.
- P. A. Fillmore, J. G. Stampfli, and J. P. Williams, On the essential numerical range, the essential spectrum, and a problem of Halmos, Acta Sci. Math. (Szeged) 33 (1972), 179–192. MR 322534
- I. C. Gohberg and M. G. Kreĭn, Theory and applications of Volterra operators in Hilbert space, Translations of Mathematical Monographs, Vol. 24, American Mathematical Society, Providence, R.I., 1970. Translated from the Russian by A. Feinstein. MR 0264447
- Paul R. Halmos, A Hilbert space problem book, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR 0208368 G. H. Hardy, J. E. Littlewood, and G. Polya, Inequalities, Cambridge Univ. Press, London, 1934.
- Kenneth Hoffman, Banach spaces of analytic functions, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR 0133008
- Yitzhak Katznelson, An introduction to harmonic analysis, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0248482
- V. B. Korotkov, On integral operators with Carleman kernels, Dokl. Akad. Nauk SSSR 165 (1965), 748–751 (Russian). MR 0209921 —, Strong integral operators, Siberian Math. J. 16 (1974), 1137-1140.
- K. Kuratowski, Topology. Vol. I, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1966. New edition, revised and augmented; Translated from the French by J. Jaworowski. MR 0217751
- George W. Mackey, Borel structure in groups and their duals, Trans. Amer. Math. Soc. 85 (1957), 134–165. MR 89999, DOI 10.1090/S0002-9947-1957-0089999-2
- H. L. Royden, Real analysis, The Macmillan Company, New York; Collier Macmillan Ltd., London, 1963. MR 0151555
- Helmut H. Schaefer, Banach lattices and positive operators, Die Grundlehren der mathematischen Wissenschaften, Band 215, Springer-Verlag, New York-Heidelberg, 1974. MR 0423039, DOI 10.1007/978-3-642-65970-6
- Robert Schatten, Norm ideals of completely continuous operators, Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), Heft 27, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1960. MR 0119112, DOI 10.1007/978-3-642-87652-3 A. R. Sourour, A note on integral operators (submitted). V. S. Sunder, Unitary orbits of integral operators, Notices Amer. Math. Soc. 24 (1977), A-639.
- Joachim Weidmann, Carlemanoperatoren, Manuscripta Math. 2 (1970), 1–38 (German, with English summary). MR 283616, DOI 10.1007/BF01168477
- J. H. Anderson and J. G. Stampfli, Commutators and compressions, Israel J. Math. 10 (1971), 433–441. MR 312312, DOI 10.1007/BF02771730 J. P. Williams, The essential numerical range, unpublished manuscript, 1971.
Additional Information
- © Copyright 1979 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 253 (1979), 339-363
- MSC: Primary 47G05; Secondary 47B38
- DOI: https://doi.org/10.1090/S0002-9947-1979-0536952-2
- MathSciNet review: 536952