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 | | · 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 well-known fact that the supremum of CN (T ) over all matrices with ||T | | ≤ 1 and minimal absolute eigenvalue r = min i=1,...,n |λ i | > 0 is the Kronecker bound 1 r n . This result is subsequently generalized by computing the corresponding supremum of (ζ − T ) −1 for any |ζ| ≤ 1. 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 socalled model matrices, i.e. matrix representations of the compressed backward shift operator on the Hardy space H 2 to a finite-dimensional invariant subspace.

H ∞ interpolation constrained by Beurling–Sobolev norms

“…The question of estimating C n, r (X, H ∞ ) asymptotically as n → ∞ and r → 1 arose in an applied context in [2,3]. It is also motivated by applications in matrix analysis to estimate functions of a contraction A acting on an n-dimensional Hilbert space, for instance the powers of A [9,10,13] or its resolvent [6,11], in terms of its spectrum. From now on, for two positive functions a and b, we say that a is dominated by b, denoted by a b, if there is a constant c > 0 such that a ≤ cb; and we say that a and b are comparable, denoted by a b, if both a b and b a.…”

Infinite-dimensional features of matrices and pseudospectra

“…As positive results on estimation of norms f (T ) , one can mention the Kreiss matrix theorem (see, for instance, [50,Section 18]) and the results by Szehr and Zarouf (see [46,47] and references therein).…”

Spectral variation bounds in hyperbolic geometry

“…This estimate is sharp for |z| = 1 but we cannot leverage it. It is also possible to improve this bound and derive (a more complicated) estimate that is optimal for z ∈ σ(A) [24]. Bounds of this type demonstrate that Blaschke products occur naturally in the context of spectral variation.…”