Conservation of the Number of Eigenvalues of Finite Dimensional and Compact Operators Inside and Outside Circle

Abstract

Let  $i_{in}(A)$ and $i_{out}(A)$ be the numbers of the  eigenvalues of a  matrix $A$   lying inside and outside the unit circle, respectively.  Let $\ti A$ be a perturbed matrix. We obtain the conditions under which $i_{in}(A)=i_{in}(\ti A)$ and $i_{out}(A)=i_{out}(\ti A)$. Our main tool is the norm estimates for resolvents of operators on the tensor product of Euclidean spaces. The results for finite matrices are particularly generalized to Hilbert-Schmidt operators.