Bounds for the spectrum of a two parameter matrix eigenvalue problem

We consider the two parameter eigenvalue problem Tjvj−λ1Aj1vj−λ2Aj2vj=0, where λj∈C; Tj, Ajk(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 A11⊗A22−A12⊗A21, where ⊗ 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.