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Ideals of sets of integral and partially integral operators. (English. Russian original) Zbl 0727.47040
Sib. Math. J. 31, No. 4, 704-708 (1990); translation from Sib. Mat. Zh. 31, No. 4(182), 207-211 (1990).
The authors shows that
(i) any right (two-sided) ideal of the set of all partially integral operators in $$L_ 2$$, or of the set of all integral operators in $$L_ 2$$, is contained in the set C $$(C_ 2)$$ of all Carleman (Hilbert- Schmidt) integral oprators in $$L_ 2;$$
(ii) any left ideal of the set of all partially integral operators in $$L_ 2$$ is contained in the left ideal of this set consisting of all continuous linear operators in $$L_ 2$$ that map the unit ball in $$L_{\infty}$$ into an ellipsoid in $$L_ 2;$$
(iii) any left ideal of the set of all integral operators in $$L_ 2$$ is contained in the left ideal of this set consisting of all regular integral operators in $$C_ 2$$.

##### MSC:
 47G10 Integral operators 47B10 Linear operators belonging to operator ideals (nuclear, $$p$$-summing, in the Schatten-von Neumann classes, etc.)
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##### References:
 [1] V. B. Korotkov, Integral Operators [in Russian], Nauka, Novosibirsk (1983). · Zbl 0526.47015 [2] V. N. Sudakov, Geometric Problems of the Theory of Infinite-Dimensional Probability Distributions [in Russian], Tr. Mat. Inst. Im. V. A. Steklova Akad. Nauk SSSR, Vol. 141 (1976). · Zbl 0409.60005 [3] P. R. Halmos and V. S. Saunder, Bounded Integral Operators on L2 Spaces, Springer-Verlag, Berlin-New York (1978). [4] N. Dunford and J. T. Schwartz, Linear Operators, Part 1, General Theory, Interscience, New York (1958). [5] A. V. Bukhvalov, ?On integral representation of linear operators,? Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR,47, 5-14 (1974).
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