Abstract
LetH be a separable Hilbert space; an operator A fromH toL 2(X,μ) is called a Carleman operator, if there exists a μ-measurable function k defined onX with values inH, such that for every fεD(A) the equation Af(x)=(f,k(x)) holds μ-almost everywhere. This is a generalization of STONE's notion of a Carleman operator [13]. In the same sense we generalize the notions of operators of Carleman type and strong Carleman operators, originally given by MISRA, SPEISER and TARGONSKI [8]. We give several characterizations of these operators for every measure space (X, μ) such thatL 2(X, μ) is separable and infinite dimensional.
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Weidmann, J. Carlemanoperatoren. Manuscripta Math 2, 1–38 (1970). https://doi.org/10.1007/BF01168477
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DOI: https://doi.org/10.1007/BF01168477