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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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##### References:
 [1] J. Arazy, S. D. Fisher, and J. Peetre, Besov norms of rational functions, Function spaces and applications (Lund, 1986) Lecture Notes in Math., vol. 1302, Springer, Berlin, 1988, pp. 125 – 129. · Zbl 0647.46034 [2] Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Springer-Verlag, Berlin-New York, 1976. Grundlehren der Mathematischen Wissenschaften, No. 223. · Zbl 0344.46071 [3] H. Bohr, A theorem concerning power series, Proc. London Math. Soc. (2) 13 (1914), 1-5. · JFM 44.0289.01 [4] I. A. Boricheva and E. M. Dyn$$^{\prime}$$kin, A nonclassical problem of free interpolation, Algebra i Analiz 4 (1992), no. 5, 45 – 90 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 4 (1993), no. 5, 871 – 908. · Zbl 0779.30021 [5] Lennart Carleson, Sets of uniqueness for functions regular in the unit circle, Acta Math. 87 (1952), 325 – 345. · Zbl 0046.30005 [6] E. M. Dyn$$^{\prime}$$kin, Free interpolation sets for Hölder classes, Mat. Sb. (N.S.) 109(151) (1979), no. 1, 107 – 128, 166 (Russian). [7] O. Èl$$^{\prime}$$-Falla, N. K. Nikol$$^{\prime}$$skiĭ, and M. Zarrabi, Estimates for resolvents in Beurling-Sobolev algebras, Algebra i Analiz 10 (1998), no. 6, 1 – 92 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 10 (1999), no. 6, 901 – 964. [8] Omar El-Fallah and Thomas Ransford, Extremal growth of powers of operators satisfying resolvent conditions of Kreiss-Ritt type, J. Funct. Anal. 196 (2002), no. 1, 135 – 154. · Zbl 1015.47002 [9] Теория матриц, Сецонд супплементед едитион. Щитх ан аппендиш бы В. Б. Лидский, Издат. ”Наука”, Мосцощ, 1966 (Руссиан). Ф. Р. Гантмачер, Матризенречнунг. ИИ. Спезиелле Фраген унд Анщендунген, Хочсчулбüчер фüр Матхематик, Бд. 37, ВЕБ Деуцчер Верлаг дер Щиссенсчафтен, Берлин, 1959 (Герман). Ф. Р. Гантмачер, Апплицатионс оф тхе тхеоры оф матрицес, Транслатед бы Ј. Л. Бреннер, щитх тхе ассистанце оф Д. Щ. Бушащ анд С. Евануса, Интерсциенце Публишерс, Инц., Нещ Ыорк; Интерсциенце Публишерс Лтд., Лондон, 1959. Ф. Р. Гантмачер, Тхе тхеоры оф матрицес. Волс. 1, 2, Транслатед бы К. А. Хирсч, Челсеа Публишинг Цо., Нещ Ыорк, 1959. [10] E. Gluskin, M. Meyer, and A. Pajor, Zeros of analytic functions and norms of inverse matrices, Israel J. Math. 87 (1994), no. 1-3, 225 – 242. · Zbl 0817.46026 [11] Gene H. Golub and Charles F. Van Loan, Matrix computations, 3rd ed., Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, MD, 1996. · Zbl 0865.65009 [12] Colin C. Graham and O. Carruth McGehee, Essays in commutative harmonic analysis, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 238, Springer-Verlag, New York-Berlin, 1979. · Zbl 0439.43001 [13] M. B. Gribov and N. K. Nikol$$^{\prime}$$skiĭ, Invariant subspaces and rational approximation, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 92 (1979), 103 – 114, 320 (Russian, with English summary). Investigations on linear operators and the theory of functions, IX. · Zbl 0433.31010 [14] Roland Hagen, Steffen Roch, and Bernd Silbermann, Spectral theory of approximation methods for convolution equations, Operator Theory: Advances and Applications, vol. 74, Birkhäuser Verlag, Basel, 1995. · Zbl 0817.65146 [15] Alfred Horn, On the eigenvalues of a matrix with prescribed singular values, Proc. Amer. Math. Soc. 5 (1954), 4 – 7. · Zbl 0055.00908 [16] Jean-Pierre Kahane, Séries de Fourier absolument convergentes, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 50, Springer-Verlag, Berlin-New York, 1970 (French). · Zbl 0195.07602 [17] V. È. Kacnel$$^{\prime}$$son and V. I. Macaev, Spectral sets for operators in a Banach space and estimates of functions of finite-dimensional operators, Teor. Funkciĭ Funkcional. Anal. i Priložen. Vyp. 3 (1966), 3 – 10 (Russian). [18] P. I. Lizorkin, Multipliers of Fourier integrals in the spaces \?_{\?,\?}, Trudy Mat. Inst. Steklov 89 (1967), 231 – 248 (Russian). · Zbl 0159.17402 [19] Albert W. Marshall and Ingram Olkin, Inequalities: theory of majorization and its applications, Mathematics in Science and Engineering, vol. 143, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. · Zbl 0437.26007 [20] F. L. Nazarov, Private communication, August 2004 (Russian); (fedja@math.msu.edu). [21] Nikolai Nikolski, In search of the invisible spectrum, Ann. Inst. Fourier (Grenoble) 49 (1999), no. 6, 1925 – 1998. · Zbl 0947.46035 [22] Nikolai K. Nikolski, Operators, functions, and systems: an easy reading. Vol. 1, Mathematical Surveys and Monographs, vol. 92, American Mathematical Society, Providence, RI, 2002. Hardy, Hankel, and Toeplitz; Translated from the French by Andreas Hartmann. Nikolai K. Nikolski, Operators, functions, and systems: an easy reading. Vol. 2, Mathematical Surveys and Monographs, vol. 93, American Mathematical Society, Providence, RI, 2002. Model operators and systems; Translated from the French by Andreas Hartmann and revised by the author. · Zbl 1007.47001 [23] Nikolai K. Nikolski, Operators, functions, and systems: an easy reading. Vol. 1, Mathematical Surveys and Monographs, vol. 92, American Mathematical Society, Providence, RI, 2002. Hardy, Hankel, and Toeplitz; Translated from the French by Andreas Hartmann. Nikolai K. Nikolski, Operators, functions, and systems: an easy reading. Vol. 2, Mathematical Surveys and Monographs, vol. 93, American Mathematical Society, Providence, RI, 2002. Model operators and systems; Translated from the French by Andreas Hartmann and revised by the author. · Zbl 1007.47001 [24] N. Nikolski, Estimates of the spectral radius and the semigroup growth bound in terms of the resolvent and weak asymptotics, Algebra i Analiz 14 (2002), no. 4, 141 – 157; English transl., St. Petersburg Math. J. 14 (2003), no. 4, 641 – 653. · Zbl 1047.47031 [25] N. K. Nikol$$^{\prime}$$skiĭ, Treatise on the shift operator, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 273, Springer-Verlag, Berlin, 1986. Spectral function theory; With an appendix by S. V. Hruščev [S. V. Khrushchëv] and V. V. Peller; Translated from the Russian by Jaak Peetre. [26] Избранные задачи весовой аппроксимации и спектрал$$^{\приме}$$ного анализа., Издат. ”Наука” Ленинград. Отдел., Ленинград, 1974 (Руссиан). Труды Мат. Инст. Стеклов. 120 (1974). Н. К. Никол$$^{\приме}$$ский, Селецтед проблемс оф щеигхтед аппрошиматион анд спецтрал аналысис, Америцан Матхематицал Социеты, Провиденце, Р.И., 1976. Транслатед фром тхе Руссиан бы Ф. А. Цезус. [27] N. K. Nikol$$^{\prime}$$skiĭ, Invariant subspaces in operator theory and function theory, Mathematical analysis, Vol. 12 (Russian), Akad. Nauk SSSR Vsesojuz. Inst. Naučn. i Tehn. Informacii, Moscow, 1974, pp. 199 – 412, 468. (loose errata) (Russian). [28] Приближение функций многих переменных и теоремы вложения, Издат. ”Наука”, Мосцощ, 1969 (Руссиан). С. М. Никол$$^{\приме}$$ский, Аппрошиматион оф фунцтионс оф северал вариаблес анд имбеддинг тхеоремс, Спрингер-Верлаг, Нещ Ыорк-Хеиделберг., 1975. Транслатед фром тхе Руссиан бы Јохн М. Данскин, Јр.; Дие Грундлехрен дер Матхематисчен Щиссенсчафтен, Банд 205. [29] Jaak Peetre, New thoughts on Besov spaces, Mathematics Department, Duke University, Durham, N.C., 1976. Duke University Mathematics Series, No. 1. · Zbl 0356.46038 [30] Vladimir V. Peller, Estimates of functions of power bounded operators on Hilbert spaces, J. Operator Theory 7 (1982), no. 2, 341 – 372. · Zbl 0485.47007 [31] -, Estimates of functions of Hilbert space operators, similarity to a contraction and related function algebras, Linear and Complex Analysis Problem Book, Lecture Notes in Math., vol. 1043, Springer-Verlag, Berlin, 1984, pp. 199-204. [32] Hervé Queffélec, Sur un théorème de Gluskin-Meyer-Pajor, C. R. Acad. Sci. Paris Sér. I Math. 317 (1993), no. 2, 155 – 158 (French, with English and French summaries). · Zbl 0810.47012 [33] Juan Jorge Schäffer, Norms and determinants of linear mappings, Math. Z. 118 (1970), 331 – 339. · Zbl 0195.41602 [34] N. A. Shirokov, Zero sets of analytic functions from the space \?^{1/\?}_{\?,1} are Carleson sets, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 113 (1981), 253 – 257, 270 (Russian, with English summary). Investigations on linear operators and the theory of functions, XI. · Zbl 0489.46027 [35] Béla Sz.-Nagy and Ciprian Foiaş, Analyse harmonique des opérateurs de l’espace de Hilbert, Masson et Cie, Paris; Akadémiai Kiadó, Budapest, 1967 (French). Béla Sz.-Nagy and Ciprian Foiaş, Harmonic analysis of operators on Hilbert space, Translated from the French and revised, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York; Akadémiai Kiadó, Budapest, 1970. [36] Hans Triebel, Spaces of Besov-Hardy-Sobolev type, BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1978. Teubner-Texte zur Mathematik; With German, French and Russian summaries. · Zbl 0408.46024 [37] Nicholas Th. Varopoulos, Some remarks on \?-algebras, Ann. Inst. Fourier (Grenoble) 22 (1972), no. 4, 1 – 11 (English, with French summary). · Zbl 0235.46074 [38] I. V. Videnskiĭ and N. A. Shirokov, On an extremal problem in the Wiener algebra, Algebra i Analiz 11 (1999), no. 6, 122 – 138 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 11 (2000), no. 6, 1035 – 1049. [39] Pascale Vitse, Functional calculus under Kreiss type conditions, Math. Nachr. 278 (2005), no. 15, 1811 – 1822. · Zbl 1099.47016 [40] Pascale Vitse, Functional calculus under the Tadmor-Ritt condition, and free interpolation by polynomials of a given degree, J. Funct. Anal. 210 (2004), no. 1, 43 – 72. · Zbl 1065.47015 [41] -, A Besov algebra functional calculus for Tadmor-Ritt operators, AAA Preprint Series, Univ. Ulm, 2004. [42] John Wermer, Potential theory, Lecture Notes in Mathematics, Vol. 408, Springer-Verlag, Berlin-New York, 1974. · Zbl 0297.31001
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