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Integral equalities for functions of unbounded spectral operators in Banach spaces

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Silvestri B.

Instytut Matematyczny Polskiej Akademii Nauk

The work is dedicated to investigating a limiting procedure for extending "local" integral operator equalities to "global" ones in the sense explained below, and to applying it to obtaining generalizations of the Newton-Leibniz formula for operator-valued functions for a wide class of unbounded operators. The integral equalities considered have the form $g(R_{F}) ∫f_{x}(R_{F})dμ(x) = h(R_{F})$. (1) They involve functions of the kind $X ∋ x ↦ f_{x}(R_{F}) ∈ B(F)$, where X is a general locally compact space, F runs over a suitable class of Banach subspaces of a fixed complex Banach space G, in particular F = G. The integrals are with respect to a general complex Radon measure on X and the $σ(B(F),𝓝_{F})$-topology on B(F), where $𝓝_{F}$ is a suitable subset of B(F)*, the topological dual of B(F). $R_{F}$ is a possibly unbounded scalar type spectral operator in F such that $σ(R_{F}) ⊆ σ(R_{G})$, and for all x ∈ X, $f_{x}$ and g,h are complex-valued Borelian maps on the spectrum $σ(R_{G})$ of $R_{G}$. If F ≠ G we call the integral equality (1) "local", while if F = G we call it "global".

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1 Silvestri B.

1 2009

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Integral equalities for functions of unbounded spectral operators in Banach spaces