On the localized weak Bishop's property (Q2192856)

Let \({\mathcal B}(X)\) be the algebra of all bounded linear operators on a complex Banach space \(X\), and \(I\) be the identity operator on \(X\). An operator \(T\in{\mathcal B}(X)\) is said to verify weak property \((\beta)\) at a complex number \(\lambda\in\mathbb{C}\) if for every open disk \(D\) centered at \(\lambda\) and every bounded sequence \((f_n)_n\) of \(X\)-valued analytic functions on \(D\) for which \((T-z)f_n(z)\) converges locally uniformly to \(0\) on \(D\), it follows that \((f_n)_n\) converges locally uniformly to \(0\) on \(D\), too. The set of all operators in \({\mathcal B}(X)\) verifying weak property \((\beta)\) at \(\lambda\) is denoted by \(w\beta(\lambda)\). Among other things, the authors show that \(w\beta(0)\) is a regularity; i.e., the following two assertions hold. \begin{enumerate} \item If \(T\in{\mathcal B}(X)\) and \(n\geq1\) is an integer, then \(T\in w\beta(0)\) if and only if \(T^n\in w\beta(0)\). \item If \(S,T,U,V\in {\mathcal B}(X)\) are mutually commuting operators such that \(SU+VT=I\), then \(ST\in w\beta(0)\) if and only if both \(S\) and \(T\) are in \(w\beta(0)\). \end{enumerate} As an immediate consequence of this and the general theory of regularities, it is then concluded that the spectrum corresponding to the regularity \(w\beta(0)\) is contained in the classical spectrum and satisfies the spectral mapping theorem. It is also shown that, if \(T\) is a bounded linear operator on a complex Hilbert space and \(T=U|T|\) is its polar decomposition, then \(T\) is in \(w\beta(0)\) if and only if so is its Aluthge transform \(\widehat{T}:=|T|^{\frac{1}{2}}U|T|^{\frac{1}{2}}\). Furthermore, the authors show that, if \(T,R\in {\mathcal B}(X)\) are commuting operators such that \(R\) is a Riesz operator, then \(T\in w\beta(0)\) if and only if \(T-R\in w\beta(0)\). The authors do not define \(\sigma_{w\beta}(.)\), but it is clear that this set is the spectrum corresponding to the regularity \(w\beta(0)\); i.e., \[\sigma_{w\beta}(T):=\{\lambda\in\mathbb{C}:T-\lambda\not\in w\beta(0)\}\] for all \(T\in{\mathcal B}(X)\).