E. B. Davies, Barry Simon
Eigenvalue Estimates for Non-normal Matrices and the Zeros of Random Orthogonal Polynomials on the Unit Circle
(292K, pdf)

ABSTRACT.  We prove that for any $n\times n$ matrix, $A$, and $z$ with $|z|\geq \|A\|$, we have that $\|(z-A)^{-1}\|\leq\cot (\frac{\pi}{4n}) \dist (z,\spec(A))^{-1}$. We apply this result to the study of random orthogonal polynomials on the unit circle.