Condition numbers of large matrices, and analytic capacities

Abstract

Given an operator T : X −→ X on a Banach space X, we compare the condition number of T ,C N(T )= � T �·� T −1 � , and the spectral condition number defined as SCN(T )= � T �· r(T −1 ), where r(·) stands for the spectral radius. For a set Y of operators, we put Φ(∆) = sup{CN(T ): T ∈ Y, SCN(T ) ≤ ∆} ,∆ ∈ (1, ∞), and say that Y is spectrally Φ-conditioned. As Y we consider certain sets of (n × n)- matrices or, more generally, algebraic operators with deg(T ) ≤ n that admit a specific functional calculus. In particular, the following sets are included: Hilbert (Banach) space power bounded matrices (operators), polynomially bounded matrices, Kreiss type matrices, Tadmor-Ritt type matrices, and matrices (operators) admitting a Besov class B s-functional calculus. The above function Φ is estimated in terms of the analytic capacity capA(·) related to the corresponding function class A.I n particular, for A = B s p,q, the quantity Φ(∆) is equivalent to ∆ n n s as ∆ −→ ∞ (or as n −→ ∞ )f or s> 0, and is bounded by ∆ n (log(n)) 1/q for s =0 .