TY - JOUR

T1 - SIMILARITY of OPERATORS on TENSOR PRODUCTS of SPACES and MATRIX DIFFERENTIAL OPERATORS

AU - Gil, Michael

N1 - Publisher Copyright:
© 2018 Australian Mathematical Publishing Association Inc.

PY - 2019/2/1

Y1 - 2019/2/1

N2 - 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 In, Iϵ 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.

AB - 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 In, Iϵ 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.

KW - matrix differential operators

KW - operator functions

KW - resolvent

KW - spectrum perturbations

KW - tensor products of operators

UR - http://www.scopus.com/inward/record.url?scp=85047847270&partnerID=8YFLogxK

U2 - 10.1017/S1446788718000058

DO - 10.1017/S1446788718000058

M3 - Article

AN - SCOPUS:85047847270

VL - 106

SP - 19

EP - 30

JO - Journal of the Australian Mathematical Society

JF - Journal of the Australian Mathematical Society

SN - 1446-7887

IS - 1

ER -