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On a problem of Halmos and Sunder. (English. Russian original) Zbl 0598.47058

Sov. Math., Dokl. 30, 384-386 (1984); translation from Dokl. Akad. Nauk SSSR 278, 296-298 (1984).
Let \({\mathcal A}\) be the collection of all integral operators in \({\mathcal B}={\mathcal B}(J^ 2(K,\mu))\). The following problem is posed in the monograph [Bounded integral operators on \(L^ 2\) spaces (1978; Zbl 0389.47001)] of P. R. Halmos and V. S. Sunder: Is \({\mathcal A}^ a \)subalgebra of the algebra \({\mathcal B}\), i.e., is the product of any two integral operators in \({\mathcal B}\) an integral operator ? This problem was solved by K. B. Korotkov [Integral operators (in Russian) (1983; Zbl 0526.47015), § 8 in Chapter 1], who constructed two integral operators whose product is not an integral operator. In this paper the Halmos-Sunder problem is considered for convolution operators. This case was not studied in [Korotkov, loc. cit.], and requires a special approach.

MSC:

47Gxx Integral, integro-differential, and pseudodifferential operators
47B38 Linear operators on function spaces (general)
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
44A35 Convolution as an integral transform
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