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Functional operators and families of set functions. (Russian) Zbl 0625.47023

Let (T,\(\Sigma\),\(\mu)\) be a measurable space with \(\sigma\)-finite measure \(\mu\) and \(\tilde X\) be a vector space of (vector-valued) “functions” with respect to the \(\mu\)-equivalence. Moreover \(\tilde X\) is supposed to be an F-space and to satisfy some natural additional conditions. To any operator f: \(\tilde X\to Z\) is associated the following family of set functions: \[ \phi_ x(e)=f(x\chi_ e),\quad e\in \Sigma,\quad x\in \tilde X, \] where \(\chi_ e\) denotes the characteristic function of e. The paper contains several statements concerning the correspondence between the properties of f and those of the associated family of set functions. No proof is given.
Reviewer: Şt.Frunză

MSC:

47B38 Linear operators on function spaces (general)
28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
47B40 Spectral operators, decomposable operators, well-bounded operators, etc.
47A70 (Generalized) eigenfunction expansions of linear operators; rigged Hilbert spaces
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