Maximum of the resolvent over matrices with given spectrum

In numerical analysis it is often necessary to estimate the condition number CN(T)=‖T‖⋅‖T−1‖ and the norm of the resolvent ‖(ζ−T)−1‖ of a given n×n matrix T. We derive new spectral estimates for these quantities and compute explicit matrices that achieve our bounds. We recover the fact that the supremum of CN(T) over all matrices with ‖T‖≤1 and minimal absolute eigenvalue r=minλ∈σ(T)⁡|λ|>0 is the Kronecker bound 1rn. This result is subsequently generalized by computing for given ζ in the closed unit disc the supremum of ‖(ζ−T)−1‖, where ‖T‖≤1 and the spectrum σ(T) of T is constrained to remain at a pseudo-hyperbolic distance of at least r∈(0,1] around ζ. 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 occurring Toeplitz matrices are so-called model matrices, i.e. matrix representations of the compressed backward shift operator on the Hardy space H2 to a finite-dimensional invariant subspace.