Some spectral properties of analytic elementary operators. (Q1396053)

Let \(\mathcal{L}(\mathfrak{X})\) denote the algebra of all bounded linear operators on a Banach space \(\mathfrak{X}\) and \(A,\;B\in \mathcal{L}(\mathfrak{X})\). An analytic elementary operator (induced by \(A,\;B\)) \(\Psi \) on \(\mathcal{L}(\mathfrak{X})\) is defined by \(\Psi (X)={\sum_{i=1}^n}f_{i}(A)Xg_{i}(B)\), where \(f_{1},\dots,f_{n}\) (resp. \(g_{1},\dots,g_{n}\)) are complex-valued functions analytic in a neighborhood of the spectrum of \(A\) (resp. \(B\)). It is well -known (Lumer-Rosenblum's theorem) that the spectrum of \(\Psi\) is given by \(\sigma (\Psi )=\{ {\sum_{i=1}^n} f_{i}(\alpha)g_{i}(\beta)\mid \alpha \in \sigma (A)\), \(\beta \in \sigma (B)\}\). In this paper, for the approximate point and defect spectra of \(\Psi\), the author gives the following parallel result to Lumer-Rosenblum's theorem \(\sigma _{\pi}(\Psi)\supseteq \{ {\sum_{i=1}^n} f_{i}(\alpha)g_{i}(\beta)\mid \alpha \in \sigma_{\pi}(A)\), \(\beta \in \sigma_{\delta }(B)\}\) and \(\sigma _{\delta }(\Psi )\supseteq \{ {\sum_{i=1}^n} f_{i}(\alpha )g_{i}(\beta )\mid \alpha \in \sigma _{\delta }(A)\), \(\beta \in \sigma _{\pi }(B)\} \), where \(\sigma _{\pi }(.)\) and \(\sigma _{\delta }(.)\) denote the approximate point and defect spectra. Moreover, in the case where \(\mathfrak{X}\) is a Hilbert space, it is proved that \(\sigma (A)=\sigma _{\delta }(A)\) and \(\sigma (B)=\sigma _{\pi }(B)\) (resp. \(\sigma (B)=\sigma _{\delta }(B)\) and \(\sigma (A)=\sigma _{\pi }(A)\)) then \(\sigma (\Psi )=\sigma _{\delta }(\Psi )=\sigma _{r}(\Psi )\) (resp. \(\sigma (\Psi )=\sigma _{\pi }(\Psi )=\sigma _{l}(\Psi )\)), where \(\sigma _{r}(.)\), \(\sigma _{l}(.)\) denote the right and left spectrum, respectively.