Maximum of the resolvent over matrices with given spectrum

Abstract

In numerical analysis it is often necessary to estimate the condition number $CN(T)=||T||_{} \cdot||T^{-1}||_{}$ and the norm of the resolvent $||(ζ-T)^{-1}||_{}$ of a given $n\times n$ matrix $T$. We derive new spectral estimates for these quantities and compute explicit matrices that achieve our bounds. We recover the well-known fact that the supremum of $CN(T)$ over all matrices with $||T||_{} \leq1$ and minimal absolute eigenvalue $r=\min_{i=1,...,n}|λ_{i}|>0$ is the Kronecker bound $\frac{1}{r^{n}}$. This result is subsequently generalized by computing the corresponding supremum of $||(ζ-T)^{-1}||_{}$ for any $|ζ| \leq1$. We find that the supremum is attained by a triangular Toeplitz matrix. This provides a simple class of structured matrices on which condition numbers and resolvent norm bounds can be studied numerically. The occuring Toeplitz matrices are so-called model matrices, i.e. matrix representations of the compressed backward shift operator on the Hardy space $H_2$ to a finite-dimensional invariant subspace.