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

Abstract

Let H=X⊗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=∑k=0mBk⊗Sk, where Bk (k=0,…,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.