On bounded local resolvents (Q2501058)

Let \(T\) be a bounded linear operator on a complex Banach space \(X\). It is known that the resolvent map \(R_T:\lambda\mapsto(T-\lambda)^{-1}\) is an unbounded and analytic on the resolvent set \(\rho(T)\) of \(T\). If \(T\) has the single-valued extension property, then, for every \(x\in X\), the map \(\lambda\mapsto(T-\lambda)^{-1}x\), which is of course analytic on \(\rho(T)\), admits a maximal analytic extension \(R_T(\cdot,x)\) satisfying \[ (T-\lambda)R_T(\lambda,x)=x \] for all \(\lambda\) in the domain of \(R_T(\cdot,x)\). This extension is called the local resolvent map of \(T\) at \(x\), and its domain coincides, in fact, with the local resolvent set \(\rho_T(x)\) of \(T\) at \(x\). Unlike the resolvent map, the local resolvent maps may be bounded. In [Integral Equations Oper.\ Theory 34, No.\,1, 1--8 (1999; Zbl 0931.47003)], \textit{T.\,Bermúdez} and \textit{M.\,González} showed that a normal operator on a separable Hilbert space has a nontrivial bounded local resolvent map if and only if the interior of its spectrum is not empty. In [Suppl.\ Rend.\ Circ.\ Mat.\ Palermo, II.\,Ser.\ 56, 15--25 (1998; Zbl 0929.47001)], \textit{M.\,M.\,Neumann} extended this result to nonseparable spaces and established a similar result for some multiplication operators. Let \(T\) be a bounded linear operator on a complex Banach space \(X\) and assume that the spectrum of \(T\) contains a nonempty open set \(U\). In the present article, the authors prove that if \(T\) has the single-valued extension property and the decomposition property \((\delta)\), then there exists \(x\in X\) such that \(\sigma_T(x):=\mathbb{C}\backslash\rho_T(x)=\overline{U}\) and the local resolvent map \(R_T(\cdot,x)\) is bounded on \(\rho_T(x)\). They also show that the same result holds if the point spectrum of \(T\) has empty interior and \(U\) is contained in the localisable spectrum of \(T\).