Pointwise functional calculi (Q2563441)

If \(A\) is a closed linear operator on a Banach space \(X\) and \({\mathcal F}\) is a Banach algebra of functions, then a pointwise \({\mathcal F}\) functional calculus for \(A\) at \(x\in X\) is ``a bounded linear map from \({\mathcal F}\) into \(X\), with the properties one would expect from a map \(f\mapsto f(A)x\), if \(A\) had a \({\mathcal F}\) functional calculus'' (in the author's words). One of the main results of the paper under review is that, under natural hypotheses, the set of all points \(x\in X\) at which \(A\) has a pointwise \({\mathcal F}\) functional calculus constitute a Banach space \(Z({\mathcal F})\) that is continuously embedded in \(X\) and such that \(A|_{Z({\mathcal F})}\) has a \({\mathcal F}\) functional calculus; this Banach space is maximal with the last two properties. The relationship among pointwise functional calculi, semigroups of operators and the abstract Cauchy problem is then investigated. Pointwise generalized scalar operators are discussed towards the end of the paper. One proves here pointwise versions of several of the basic properties of generalized scalar operators [cf. \textit{S. Kantorovitz}, ``Spectral theory of Banach space operators. \(C_k\)-classification, abstract Volterra operators, similarity, spectrality, local spectral analysis''. Lect. Notes Math. 1012 (1983; Zbl 0527.47001); see also \textit{I. Colojoară} and \textit{C. Foiaş}, ``Theory of generalized spectral operators'', New York (1968; Zbl 0189.44201)]. All of these results are beautifully illustrated by examples of weighted shifts on sequence spaces and weighted translations on function spaces.