On the stability of the localized single-valued extension property under commuting perturbations (Q2838960)

A bounded linear operator \(T\) acting on a complex Banach space \(X\) is said to have the single-valued extension property (SVEP) at a point \(\lambda\in\mathbb{C}\) if, for every open disc \(D\subset\mathbb{C}\) centered at \(\lambda\), the only analytic function \(f\,:\,D\rightarrow X\) that satisfies \((T-\mu I)f(\mu)=0\) for all \(\mu\in D\) is the function \(f\equiv 0\). When \(T\) has the SVEP at every point \(\lambda\in\mathbb{C}\), then \(T\) is said to have the SVEP. In general, the SVEP is not stable under sums and products of commuting operators, as proved by \textit{A. Bourhim} and \textit{V. G. Miller} [Glasg. Math. J. 49, No. 1, 99--104 (2007; Zbl 1122.47032)] using the theory of weighted shift operators. The paper under review presents a general principle showing that counterexamples exist in abundance. They also investigate the stability of the SVEP under commuting perturbation by nilpotent, quasi-nilpotent, algebraic or Riesz operators.NEWLINENEWLINENEWLINERemark. The reviewer would like to point out that the result of Theorem 2.2 can be deduced immediately from Lemma 2.8 of [\textit{C. Benhida} et al., Acta Sci. Math. 71, No. 3--4, 681--690 (2005; Zbl 1105.47005)].