Estimates for resolvents and functions of operator pencils on tensor products of Hilbert spaces
Let \(X, Y\) be separable Hilbert spaces. The author establishes some norm estimates for the resolvent and operator-valued functions of the operator \(A =\sum_{k=0}^m B_k\otimes S_k\), where \(B_k\,\,(k =0, \dots , m)\) are bounded operators acting on \(Y\) and \(S\) is an invertible selfadjoint operator acting in \(X\). By these estimates, he investigates spectrum perturbations of the operator pencil \(A\). Some improvements are given in the case where \(Y=\mathbb{C}^n\). The author then uses his results to study the non-selfadjoint differential operators with matrix and operator coefficients in Hilbert spaces. The reader may also want to consult \textit{L. A. Shuster} [J. Math. Anal. Appl. 229, No.~2, 363--375 (1999; Zbl 0919.34070)], where the problem of localization of the spectrum for a class of semiregular boundary value problems is investigated.
scientific article; zbMATH DE number 5960767
scientific article