Integral equalities for functions of unbounded spectral operators in Banach spaces

Benedetto Silvestri. Integral equalities for functions of unbounded spectral operators in Banach spaces. 2009. <http://eudml.org/doc/286000>.

@book{BenedettoSilvestri2009,
abstract = {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 $σ(B(F),_\{F\})$-topology on B(F), where $_\{F\}$ 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”.},
author = {Benedetto Silvestri},
keywords = {unbounded spectral operators in Banach spaces; functional calculus; integration of locally convex space valued maps},
language = {eng},
title = {Integral equalities for functions of unbounded spectral operators in Banach spaces},
url = {http://eudml.org/doc/286000},
year = {2009},
}

TY - BOOK
AU - Benedetto Silvestri
TI - Integral equalities for functions of unbounded spectral operators in Banach spaces
PY - 2009
AB - 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 $σ(B(F),_{F})$-topology on B(F), where $_{F}$ 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”.
LA - eng
KW - unbounded spectral operators in Banach spaces; functional calculus; integration of locally convex space valued maps
UR - http://eudml.org/doc/286000
ER -