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Integral equalities for functions of unbounded spectral operators in Banach spaces. (English) Zbl 1235.47035

The author studies a limiting procedure for extending “local” integral operators to “global” ones and applies it to obtain a generalization of the Newton-Leibniz formula for operator-valued functions to unbounded scalar type spectral operators. The integral equalities considered have the form \[ g(R_F)\int f_x(R_F )\,d\mu(x)=h(R_F). \] They involve functions of the kind \(X\ni x\mapsto f_x(R(R_F))\in B(F)\), where \(X\) is a locally compact spaces, \(F\) runs over a suitable class of Banach subspaces of a fixed complex Banach space \(G\), in particular \(F=G\), \(R_F\) is a possibly unbounded scalar type spectral operator in \(F\) such that \(\sigma(R_F)\subset \sigma(R_G)\), and for all \(x\in X\), \(f_x, g\) and \(h\) are complex valued Bore maps on the spectrum \(\sigma(R_G)\) of \(R_G\).
The main result of this paper, Theorem 2.25, gives a condition under which the equation \[ g(R|E(\sigma_n )G)\int f_x (R|E(\sigma_n )G)\,d\mu(x)=h(E(\sigma_n )G)\in B(E(\sigma_n )G) \] implies that \(g(R)\int f_x(R)\,d\mu(x)=h(G)\in B(G)\).

MSC:

47B40 Spectral operators, decomposable operators, well-bounded operators, etc.
46G10 Vector-valued measures and integration
47A60 Functional calculus for linear operators
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