Axiomatic theory of spectrum. III: Semiregularities (Q2703083)

[For part II see \textit{M. Mbekhta} and the author, ibid. 119, No. 2, 129-147 (1996; Zbl 0857.47002).]NEWLINENEWLINENEWLINEThe author defines the notions of upper and lower semiregularities in Banach algebra and proves one way spectral mapping theorems.NEWLINENEWLINENEWLINEDefinition 1: Let \({\mathfrak R}\) be a non-empty subset of a Banach algebra \(A\). Then \({\mathfrak R}\) is called a lower semiregular ifNEWLINENEWLINENEWLINE(i) \(a\in A\), \(n\in\aleph\), \(a^n\in{\mathfrak R}\Rightarrow a\in{\mathfrak R}\),NEWLINENEWLINENEWLINE(ii) if \(a\), \(b\), \(c\), \(d\) are mutually commuting elements of \(A\) satisfying \(ac+ bd=1_A\) and \(ab\in{\mathfrak R}\) then \(a,b\in{\mathfrak R}\).NEWLINENEWLINENEWLINEThe corresponding spectrum \(\sigma_{{\mathfrak R}}\) is defined by \(\sigma_{{\mathfrak R}}= \{\lambda\in C: a-\lambda\not\in{\mathfrak R}\}\). Regarding this notion the author proves the following main result:NEWLINENEWLINENEWLINETheorem 1: Let \({\mathfrak R}\subset A\) be a lower semiregularity and \(a\in A\). Then \(f(\sigma_{{\mathfrak R}}(a))\subset \sigma_{{\mathfrak R}}(f(a))\) for each locally non-constant function \(f\) analytic on a neighbourhood of \(\sigma(a)\).NEWLINENEWLINENEWLINEA similar definition and result is given for the notion of upper semiregularity. The article contains many more results.