SIMILARITY of OPERATORS on TENSOR PRODUCTS of SPACES and MATRIX DIFFERENTIAL OPERATORS

Let {equation presented} be the tensor product of a Euclidean space Cⁿ and a separable Hilbert space ϵ. Our main object is the operator {equation presented}, where S is a normal operator in ϵ, A is an n × n matrix, and I_n, I_ϵ are the unit operators in Cⁿ and ϵ, respectively. Numerous differential operators with constant matrix coefficients are examples of operator G. In the present paper we show that G is similar to an operator {equation presented} where D is a block matrix, each block of which has a unique eigenvalue. We also obtain a bound for the condition number. That bound enables us to establish norm estimates for functions of G, nonregular on the closed convex hull co(G) of the spectrum of G. The functions {equation presented} are examples of such functions. In addition, in the appropriate situations we improve the previously published estimates for the resolvent and functions of G regular on co(G). Since differential operators with variable coefficients often can be considered as perturbations of operators with constant coefficients, the results mentioned above give us estimates for functions and bounds for the spectra of differential operators with variable coefficients.