Estimates for resolvents and functions of operator pencils on tensor products of Hilbert spaces

Abstract

Let $H=X\otimes Y$ be a tensor product of separable Hilbert spaces $X$ and $Y$. We establish 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 in $Y$, and $S$ is a self-adjoint operator acting in $X$. By these estimates we investigate spectrum perturbations of $A$. The abstract results are applied to the nonself-adjoint differential operators in Hilbert and Euclidean spaces. Our main tool is a combined use of some properties of operators on tensor products of Hilbert spaces and the recent estimates for the norm of the resolvent of a nonself-adjoint operator.