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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



Condition numbers of large matrices, and analytic capacities

Author: N. K. Nikolski
Original publication: Algebra i Analiz, tom 17 (2005), nomer 4.
Journal: St. Petersburg Math. J. 17 (2006), 641-682
MSC (2000): Primary 47A60, 65F35, 15A12; Secondary 15A60, 32A38, 46J15
Published electronically: May 3, 2006
MathSciNet review: 2173939
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Abstract: Given an operator $ T:X\longrightarrow X$ on a Banach space $ X$, we compare the condition number of $ T$, $ \operatorname{CN}(T)= \Vert T\Vert \cdot \Vert T^{-1}\Vert $, and the spectral condition number defined as $ \operatorname{SCN}(T)= \Vert T\Vert \cdot r(T^{-1})$, where $ r(\cdot )$ stands for the spectral radius. For a set $ \Upsilon$ of operators, we put $ \Phi (\Delta) = \sup\{\operatorname{CN}(T): T\in \Upsilon , \operatorname{SCN}(T) \le \Delta \}$, $ \Delta \in [1,\infty )$, and say that $ \Upsilon $ is spectrally $ \Phi $-conditioned. As $ \Upsilon $ we consider certain sets of $ (n\times n)$-matrices or, more generally, algebraic operators with $ \deg(T)\le 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}_{p,q}$-functional calculus. The above function $ \Phi $ is estimated in terms of the analytic capacity $ \operatorname{cap}_{A}(\cdot )$ related to the corresponding function class $ A$. In particular, for $ A= B^{s}_{p,q}$, the quantity $ \Phi (\Delta )$ is equivalent to $ \Delta ^{n}n^{s}$ as $ \Delta \longrightarrow \infty $ (or as $ n\longrightarrow \infty $) for $ s>0$, and is bounded by $ \Delta ^{n}(\log(n))^{1/q}$ for $ s=0$.

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Additional Information

N. K. Nikolski
Affiliation: Département de Mathématiques, Université de Bordeaux 1, 351, cours de la Libération, 33405 Talence, France, and St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia

Keywords: Condition numbers, algebraic operators, function algebras, Wiener algebra, Besov spaces, capacity
Received by editor(s): April 15, 2005
Published electronically: May 3, 2006
Article copyright: © Copyright 2006 American Mathematical Society

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