Remote Access St. Petersburg Mathematical Journal

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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$.

References [Enhancements On Off] (What's this?)

  • [AFP] 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. MR 942262, 10.1007/BFb0078868
  • [BL] Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Springer-Verlag, Berlin-New York, 1976. Grundlehren der Mathematischen Wissenschaften, No. 223. MR 0482275
  • [Bo] H. Bohr, A theorem concerning power series, Proc. London Math. Soc. (2) 13 (1914), 1-5.
  • [BD] I. A. Boricheva and E. M. Dyn′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. MR 1202724
  • [C] Lennart Carleson, Sets of uniqueness for functions regular in the unit circle, Acta Math. 87 (1952), 325–345. MR 0050011
  • [D] E. M. Dyn′kin, Free interpolation sets for Hölder classes, Mat. Sb. (N.S.) 109(151) (1979), no. 1, 107–128, 166 (Russian). MR 538552
  • [ENZ] O. Èl′-Falla, N. K. Nikol′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. MR 1678988
  • [ER] 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. MR 1941994, 10.1006/jfan.2002.3934
  • [G] F. R. Gantmaher, Teoriya matrits, Second supplemented edition. With an appendix by V. B. Lidskiĭ, Izdat. “Nauka”, Moscow, 1966 (Russian). MR 0202725
    F. R. Gantmacher, The theory of matrices. Vols. 1, 2, Translated by K. A. Hirsch, Chelsea Publishing Co., New York, 1959. MR 0107649
  • [GMP] 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. MR 1286828, 10.1007/BF02772996
  • [GVL] 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. MR 1417720
  • [GrMcG] 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. MR 550606
  • [GrN] M. B. Gribov and N. K. Nikol′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. MR 566744
  • [HRS] 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. MR 1320262
  • [Hor] Alfred Horn, On the eigenvalues of a matrix with prescribed singular values, Proc. Amer. Math. Soc. 5 (1954), 4–7. MR 0061573, 10.1090/S0002-9939-1954-0061573-6
  • [K] Jean-Pierre Kahane, Séries de Fourier absolument convergentes, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 50, Springer-Verlag, Berlin-New York, 1970 (French). MR 0275043
  • [KM] V. È. Kacnel′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). MR 0206715
  • [L] P. I. Lizorkin, Multipliers of Fourier integrals in the spaces 𝐿_{𝑝,𝜃}, Trudy Mat. Inst. Steklov 89 (1967), 231–248 (Russian). MR 0217519
  • [MO] 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. MR 552278
  • [Na] F. L. Nazarov, Private communication, August 2004 (Russian); (
  • [N1] Nikolai Nikolski, In search of the invisible spectrum, Ann. Inst. Fourier (Grenoble) 49 (1999), no. 6, 1925–1998. MR 1738071
  • [N2] 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. MR 1864396
  • [N3] 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. MR 1892647
  • [N4] 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. MR 1935921
  • [N5] N. K. Nikol′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. MR 827223
  • [N6] N. K. \cyr{N}ikol′skiĭ, Izbrannye zadachi vesovoi approksimatsii i spektralnogo analiza, Izdat. “Nauka” Leningrad. Otdel., Leningrad, 1974 (Russian). Trudy Mat. Inst. Steklov. 120 (1974). MR 0467269
  • [N7] N. K. Nikol′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). MR 0430821
  • [S.Ni] S. M. \cyr{N}ikol′skiĭ, Priblizhenie funktsii mnogikh peremennykh i teoremy vlozheniya, Izdat. “Nauka”, Moscow, 1969 (Russian). MR 0310616
    S. M. Nikol′skiĭ, Approximation of functions of several variables and imbedding theorems, Springer-Verlag, New York-Heidelberg., 1975. Translated from the Russian by John M. Danskin, Jr.; Die Grundlehren der Mathematischen Wissenschaften, Band 205. MR 0374877
  • [P] Jaak Peetre, New thoughts on Besov spaces, Mathematics Department, Duke University, Durham, N.C., 1976. Duke University Mathematics Series, No. 1. MR 0461123
  • [Pe1] Vladimir V. Peller, Estimates of functions of power bounded operators on Hilbert spaces, J. Operator Theory 7 (1982), no. 2, 341–372. MR 658618
  • [Pe2] -, 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.
  • [Q] 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). MR 1231413
  • [Sch] Juan Jorge Schäffer, Norms and determinants of linear mappings, Math. Z. 118 (1970), 331–339. MR 0281730
  • [Sh] N. A. Shirokov, Zero sets of analytic functions from the space 𝐵^{1/𝑝}_{𝑝,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. MR 629851
  • [SzNF] Béla Sz.-Nagy and Ciprian Foiaș, Analyse harmonique des opérateurs de l’espace de Hilbert, Masson et Cie, Paris; Akadémiai Kiadó, Budapest, 1967 (French). MR 0225183
    Béla Sz.-Nagy and Ciprian Foiaș, 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. MR 0275190
  • [T] 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. MR 581907
  • [Va] Nicholas Th. Varopoulos, Some remarks on 𝑄-algebras, Ann. Inst. Fourier (Grenoble) 22 (1972), no. 4, 1–11 (English, with French summary). MR 0338780
  • [VSh] 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. MR 1746071
  • [Vi1] Pascale Vitse, Functional calculus under Kreiss type conditions, Math. Nachr. 278 (2005), no. 15, 1811–1822. MR 2182092, 10.1002/mana.200310341
  • [Vi2] 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. MR 2051632, 10.1016/j.jfa.2003.08.002
  • [Vi3] -, A Besov algebra functional calculus for Tadmor-Ritt operators, AAA Preprint Series, Univ. Ulm, 2004.
  • [W] John Wermer, Potential theory, Lecture Notes in Mathematics, Vol. 408, Springer-Verlag, Berlin-New York, 1974. MR 0454033

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2000): 47A60, 65F35, 15A12, 15A60, 32A38, 46J15

Retrieve articles in all journals with MSC (2000): 47A60, 65F35, 15A12, 15A60, 32A38, 46J15

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