Silvestri, Benedetto Integral equalities for functions of unbounded spectral operators in Banach spaces. (English) Zbl 1235.47035 Diss. Math. 464, 60 p. (2009). 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 \(g(R)\int f_x(R)\,d\mu(x)=h(G)\in B(G)\). Reviewer: Mohammad Bagher Ghaemi (Tehran) Cited in 5 Documents MSC: 47B40 Spectral operators, decomposable operators, well-bounded operators, etc. 46G10 Vector-valued measures and integration 47A60 Functional calculus for linear operators Keywords:unbounded spectral operators in Banach spaces; functional calculus; integration of locally convex space valued maps PDFBibTeX XMLCite \textit{B. Silvestri}, Diss. Math. 464, 60 p. (2009; Zbl 1235.47035) Full Text: DOI arXiv