Bounds for the spectrum of a two parameter matrix eigenvalue problem

We consider the two parameter eigenvalue problem T(j)v(j) - lambda(1)A(j1)v(j) - lambda(2)A(j2)v(j) = 0, where lambda(j) is an element of C; T-j, A(jk) (j, k = 1,2) are matrices. Bounds for the spectral radius of that problem are suggested. Our main tool is a norm estimate for the operator inverse to the operator A(11) circle times A(22) - A(12) circle times A(21), where circle times means the tensor product. In addition, by virtue of that norm estimate and the Ostrowsky-Schneider theorem we establish a condition that provides the conservation of the number of the eigenvalues of the considered problem in a half-plane. (C) 2015 Elsevier Inc. All rights reserved.