## Integral equalities for functions of unbounded spectral operators in Banach spaces

### Volume 464 / 2009

#### Abstract

\def\n{{\cal N}}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 $$ g(R_{F})\int f_{x}(R_{F})\,d\mu(x)=h(R_{F}). \eqno(1)$$ They involve functions of the kind $$ X\ni x\mapsto f_{x}(R_{F})\in B(F), $$ 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 $\sigma(B(F),\n_{F})$-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 $\sigma(R_{F})\subseteq\sigma(R_{G})$, and for all $x\in X$, $f_{x}$ and $g,h$ are complex-valued Borelian maps on the spectrum $\sigma(R_{G})$ of $R_{G}$. If $F\neq G$ we call the integral equality (1) “local”, while if $F=G$ we call it “global”.