Localization of eigenvalues of closed linear operators
The main result of this paper is a localization theorem (Theorem 1.1) for the eigenvalues of a closed linear operator on a Banach space. Let T be such an operator and \(T_ n\) be a bounded operator. Suppose \(\Gamma\) is a contour containing inside m eigenvalues of T (counting the multiplicities) and, it must be added, no other points of the spectrum of T. If \(\sup_{\zeta \in \Gamma}Spr(R(\zeta)(T_ n-\zeta I)-I)<1,\) where Spr denotes the spectral radius and \(R(\zeta)=(T-\zeta I)^{-1},\) then \(T_ n\) has also inside of \(\Gamma\) m eigenvalues (counting the multiplicities) and no other points of the spectrum. A simple application is the well-known theorem of Gershgorin of localization of the eigenvalues of a complex \(n\times n\) matrix. Usually the operator \(T_ n\) is a finite-dimensional discretization of T and Theorem 1.1 may be used to get both apriori and aposteriori estimation of the eigenvalues of T.
closed linear operator on a Banach space
scientific article; zbMATH DE number 3863893