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Condition numbers of large matrices, and analytic capacities. (English) Zbl 1098.15002
St. Petersbg. Math. J. 17, No. 4, 641-682 (2006) and Algebra Anal. 17, No. 4, 125-180 (2005).
A problem of finding a function \(\Phi_n\) such that \(\| T^{-1}\|\leq \Phi_n(\delta)\), where \(T\) is, e. g., a set of invertible \((n \times n)\) matrices and \(\delta\) stands for the minimum modulus of the eigenvalues of \(T\), \(\delta = \min | \lambda_i (T)| \), is discussed. In numerical analysis, the usual normalization \(\| T^{-1}\| \) is replaced by another kind of normalization conditions. Here, considering the operators \(T: X \rightarrow X\) acting on a finite-dimensional Banach/Hilbert space \(X\), \(\dim X = N < \infty \), the condition number of \(T\), \(\text{CN}(T) = \|T\| \cdot \| T^{-1}\|\) and the spectral condition number \({SCN}(T) = \| T\|\cdot r(T^{-1})\), where \(r(\cdot)\) means the spectral radius, are compared.
In order to measure the size of inverses and condition numbers of a set of operators \(\Upsilon\) the function \(\Phi(\Delta)= \sup\{ \text{CN}(T): T \in \Upsilon\), \(\text{SCN} (T) \leq \Delta \}\), \(\Delta \in [1, \infty)\), is introduced and then the set \(\Upsilon\) is said to be spectrally \(\Phi\)-conditioned. The bounding function \(\Phi(\Delta)\) for sets of \((n \times n)\) matrices and algebraic operators \(\Upsilon\) with \(\deg(T) \leq n\) satisfying a specific functional calculus is estimated in terms of the analytic capacity \(\text{cap}_A(\cdot)\) related to the corresponding function space \(A\).
As the set \(\Upsilon\) the following cases are considered: the set of Hilbert space power bounded operators, the Banach space Tadmor-Ritt operators, Banach space Kreiss operators and operators allowing a Besov class \(B^{s}_{p,q}\)-functional calculus, where \(s \geq 0\), \(1 \leq p,q \leq \infty\). For the space \(A=B^{s}_{p,q}\), the value \(\Phi(\Delta)\) is equivalent to \(\Delta^n n^s\) as \(\Delta \rightarrow \infty\) (or \(n \rightarrow \infty\)) for \(s > 0\) and is bounded by \(\Delta^n (\log(n))^{1/q}\) for \(s = 0\).

MSC:
15A12 Conditioning of matrices
47A60 Functional calculus for linear operators
65F35 Numerical computation of matrix norms, conditioning, scaling
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
32A38 Algebras of holomorphic functions of several complex variables
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
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