Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

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

Request Permissions   Purchase Content 
 
 

 

Sublinear dimension growth in the Kreiss Matrix Theorem


Author: N. Nikolski
Original publication: Algebra i Analiz, tom 25 (2013), nomer 3.
Journal: St. Petersburg Math. J. 25 (2014), 361-396
MSC (2010): Primary 47A10
DOI: https://doi.org/10.1090/S1061-0022-2014-01295-2
Published electronically: May 16, 2014
MathSciNet review: 3184597
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A possible sublinear dimension growth in the Kreiss Matrix Theorem, bounding the stability constant in terms of the Kreiss resolvent characteristic, is discussed. Such a growth is proved for matrices having unimodular spectrum and acting on a uniformly convex Banach space. The principal ingredients to results obtained come from geometric properties of eigenvectors, where the approaches by C. A. McCarthy-J. Schwartz (1965) and V. I. Gurarii-N. I. Gurarii (1971) are used and compared. The sharpness issue is verified via finite Muckenhoupt bases (by using mostly the approach by M. Spijker, S. Tracogna, and B. Welfert (2003)).


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

  • [ABHN2011] W. Arendt, C. Batty, M. Hieber, and F. Neubrander, Vector-valued Laplace transforms and Cauchy problems, Second edition, Monographs in Math., vol. 96, Birkhäuser, Basel, 2011. MR 2798103 (2012b:47109)
  • [Ba2003] N. Yu. Bakaev, Constant size control in stability estimates under some resolvent conditions, Vychislit. metody i program. 4 (2003), no. 2, 348-357. (Russian)
  • [BaZ2013] A. Baranov and R. Zarouf, A model space approach to some classical inequalities for rational functions (to appear).
  • [BNi1999] N. Benamara and N. Nikolski, Resolvent tests for similarity to a normal operator, Proc. London Math. Soc. (3) 78 (1999), no. 3, 585-626. MR 1674839 (2000c:47006)
  • [Be1926] S. N. Bernstein, Lecons sur les Propriétés Extrémales et la Meilleure Approximation des Fonctions Analytiques d'une Variable Réelle, Gauthier-Villars, Paris, 1926.
  • [BoD1971] F. F. Bonsall and J. Duncan, Numerical ranges of operators on normed spaces and of elements of normed algebras, Cambridge Univ. Press, Cambridge, 1971. MR 0288583 (44:5779)
  • [BDS2000] N. Borovykh, D. Drissi, and M. N. Spijker, A note about Ritt's condition, related resolvent conditions and power bounded operators, Numer. Funct. Anal. Optim. 21 (2000), no. 3-4, 425-438. MR 1769884 (2001f:47119)
  • [BoE1995] P. Borwein and T. Erdélyi, Polynomials and polynomial inequalities, Graduate Text in Math., vol. 161, Springer-Verlag, New York, 1995. MR 1367960 (97e:41001)
  • [BoE1996] -, Sharp extensions of Bernstein's inequality to rational spaces, Mathematika 43 (1996), no. 2, 413-423. MR 1433285 (97k:26014)
  • [Cr1970] M. J. Crabb, The power inequality on normed spaces, Proc. Edinburgh Math. Soc.(2) 17 (1970), 237-240. MR 0358386 (50:10852)
  • [Do1978] E. P. Dolženko, Some sharp integral estimates of the derivatives of rational and algebraic function. Applications, Anal. Math. 4 (1978), no. 4, 247-268. (Russian) MR 524439 (81d:41017)
  • [DoD1987] E. P. Dolzenko and V. I. Danchenko, The mapping of sets of finite $ \alpha $-measure by means of rational functions, Izv. Akad. Nauk SSSR Ser. Mat. 51 (1987), no. 6, 1309-1321; English transl., Math. USSR-Izv. 31 (1988), no. 3, 621-633. MR 933966 (89f:30014)
  • [DKS1993] J. L. M. van Dorsselaer, J. F. B. M. Kraaijevanger, and M. N. Spijker, Linear stability analysis in the numerical solution of initial value problems, Acta Numerica, Cambridge Univ. Press, Cambridge, 1993, 199-237. MR 1224683 (94e:65051)
  • [Do1958] Y. Domar, On the existence of a largest subharmonic minorant of a given function, Ark. Mat. 3 (1957), no. 5, 429-440. MR 00887767 (19:408c)
  • [ElR2002] O. El-Fallah and T. 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 (2004c:47024)
  • [GMcG1979] C. C. Graham and O. C. McGehee, Essays in commutative harmonic analysis, Grundlehren Math. Wiss., vol. 238, Springer, NY-Heidelberg, 1979. MR 550606 (81d:43001)
  • [Gu1971] N. I. Gurariĭ, The coefficient sequences of basis expansions in Hilbert and Banach spaces, Izv. Akad. Nauk SSSR Ser. Math. 35 (1971), no. 1, 216-223. (Russian) MR 0278046 (43:3778)
  • [GuG1971] V. I. Gurariĭ and N. I. Gurariĭ, Bases in uniformly convex and uniformly smooth Banach spaces, Izv. Akad. Nauk SSSR Ser. Math. 35 (1971), no. 1, 210-215. (Russian) MR 0283549 (44:780)
  • [Ha1955] O. Hanner, On the uniform convexity of $ L^p$ and $ l^p$, Ark. Mat. 3 (1956), no. 19, 239-244. MR 0077087 (17:987d)
  • [Ja1972] R. C. James, Super-reflexive spaces with bases, Pacific J. Math. 41 (1972), no. 2, 409-419. MR 0308752 (46:7866)
  • [JLMR1998] R. Jones, Xin Li, R. N. Mohapatra, and R. S. Rodriguez, On the Bernstein inequality for rational functions with a prescribed zero, J. Approx. Theory 95 (1998), no. 3, 476-496. MR 1657695 (99m:41019)
  • [Kra1994] J. F. B. M. Kraaijevanger, Two counterexamples related to the Kreiss matrix theorem, BIT 34 (1994), no. 1, 113-119. MR 1429692 (98c:65154)
  • [Kr1962] H.-O. Kreiss, Über die Stabilitätsdefinition für Differenzengleichungen die partielle Differentialgleichungen approximieren, Nordisk Tidskr. Information-Behandling 2 (1962), 153-181. MR 0165712 (29:2992)
  • [KrV1977] I. È. Verbickiĭ and N. Ya. Krupnik, Sharp constants in theorems of K. I. Babenko and B. V. Hvedelidze on the boundedness of a singular operator, Sakharth. SSR Mecn. Akad. Moambe 85 (1977), no. 1, 21-24. (Russian) MR 0458257 (56:16460)
  • [La1975] G. I. Laptev, Conditions for the uniform well-posedness of the Cauchy-problem for systems of equations, Dokl. Akad. Nauk SSSR 220 (1975), no. 2, 281-284. (Russian) MR 0402257 (53:6078)
  • [LeT1984] R. J. LeVeque and L. N. Trefethen, On the resolvent condition in the Kreiss matrix theorem, BIT 24 (1984), no. 4, 584-591. MR 764830 (86c:39004)
  • [Lev1940] N. Levinson, Gap and density theorems, AMS Colloquium Publ., vol. 26, Amer. Math. Soc., Providence, 1940. MR 0003208 (2:180d)
  • [LiTz1977] J. Lindenstrauss and L. Tzafriri, Classical Banach spaces. V. I, Seguence spaces, Springer-Verlag, Berlin etc., 1977. MR 0415253 (54:3344)
  • [Lyu1999] Yu. Lyubich, Spectral localization, power boundedness and invariant subspaces under Ritt's type condition, Studia Math. 134 (1999), no. 2, 153-167. MR 1688223 (2000a:47011)
  • [McCS1965] C. A. McCarthy and J. T. Schwartz, On the norm of a finite Boolean algebra of projections, and applications to theorems of Kreiss and Morton, Comm. Pure Appl. Math. 18 (1965), 191-201. MR 0180867 (31:5097)
  • [MiS1966] J. Miller and G. Strang, Matrix theorems for partial differential and difference equations, Math. Scand. 18 (1966), 113-133. MR 0209308 (35:206)
  • [Mo1964] K. W. Morton, On a matrix theorem due to H.-O. Kreiss, Comm. Pure Appl. Math. 17 (1964), 375-379. MR 0170460 (30:698)
  • [NaZ1999] B. Nagy and J. Zemanek, A resolvent condition implying power boundedness, Studia Math. 134 (1999), no. 2, 143-151. MR 1688222 (2000g:47004)
  • [Ni2002] N. Nikolski, Operators, functions, and systems: an easy reading. Vol. 2, Model operators and systems, Amer. Math. Soc., Providence, RI, 2002. MR 1892647 (2003i:47001b)
  • [Ni2005] -, Condition numbers of large matrices, and analytic capacities, Algebra i Analiz 17 (2005), no. 4, 125-180; English transl., St. Petersburg Math. J. 17 (2006), no. 4, 641-682. MR 2173939 (2006k:15017)
  • [NiK1987] N. Nikol'skiĭ and S. Khrushchëv, A functional model and some problems of the spectral theory of functions, Tr. Mat. Inst. Steklov. 176 (1987), 97-210; English transl., Proc. Steklov Inst. Math. 1988, no. 3, 101-214. MR 902373 (88j:47012)
  • [Pe2001] A. A. Pekarskiĭ, Rational approximations of functions with derivatives in a V. I. Smirnov space, Algebra i Analiz 13 (2001), no. 2, 165-190; English transl., St. Petersburg Math. J. 13 (2002), no. 2, 281-30. MR 1834865 (2002d:41020)
  • [Pe2004] -, Bernstain-type inequalities for the derivatives of rational functions in the spaces $ L^p$, $ 0<p<1$ on Lavrent'ev curves, Algebra i Analiz 16 (2004), no. 3, 143-170; English transl. St. Petersburg Math. J. 16 (2005), no. 3, 541-560. MR 2083568 (2005f:30073)
  • [PetP1987] P. P. Petrushev and V. A. Popov, Rational approximation of real functions, Encyclopedia of Mathematics and its Applications, vol. 28, Cambridge Univ. Press, Cambridge, 1987. MR 940242 (89i:41022)
  • [Pi2011] G. Pisier, Martingales in Banach Spaces (in connection with type and cotype), Course IHP, Feb., 2-8, 2011; people.math.jussieu.fr/~pisier/ihp-pisier.pdf
  • [ReT1992] S. C. Reddy and L. N. Trefethen, Stability of the method of lines, Numer. Math. 62 (1992), no. 2, 235-267. MR 1165912 (93d:65086)
  • [Sm1985] J. C. Smith, Problems and solutions: Solutions of Advanced Problems, Amer. Math. Monthly 92 (1985), no. 10, 740-741. MR 1540768
  • [Sp1991] M. N. Spijker, On a conjecture by LeVeque and Trefethen related to the Kreiss matrix theorem BIT 31 (1991), no. 3, 551-555. MR 1127492 (92h:15012)
  • [Sp1998] -, Numerical stability. Stability estimates and resolvent conditions in the numerical solution of initial value problems, Lecture Notes, Leiden, 1998. http:// www.math.leidenuniv.nl/~spijker/
  • [STW2003] M. N. Spijker, S. Tracogna, and B. Welfert, About the sharpness of the stability estimates in the Kreiss matrix theorem, Math. Comp. 72 (2003), no. 242, 697-713. (electronic) MR 1954963 (2004b:15054)
  • [StW1997] J. C. Strikwerda and B. A. Wade, A survey of the Kreiss matrix theorem for power bounded families of matrices and its extensions, Banach Center Publ., vol. 38, Polish Acad. Sci., Warsaw, 1997. MR 1457017 (98f:15020)
  • [SzN1947] B. Szőkefalvi-Nagy, On uniformly bounded linear transformations in Hilbert space, Acta Sci. Math. (Szeged) 11 (1947), 152-157. MR 0022309 (9:191b)
  • [Ta1986] E. Tadmor, The resolvent condition and uniform power boundedness, Linear Algebra Appl. 80 (1986), 250-252.
  • [ToT1999] K.-Ch. Toh and L. N. Trefethen, The Kreiss matrix theorem on a general complex domain, SIAM J. Matrix Anal. Appl. 21 (1999), no. 1, 145-165. MR 1709731 (2000h:65054)
  • [TrE2005] L. N. Trefethen and M. Embree, Spectra and pseudospectra: The behavior of nonnormal matrices and operators, Princeton Univ. Press, Princeton, 2005. MR 2155029 (2006d:15001)
  • [Vi2004] P. Vitse, The Riesz turndown collar theorem giving an asymptotic estimate of the powers of an operator under the Ritt condition, Rend. Circ. Mat. Palermo (2) 53 (2004), no. 2, 283-312. MR 2078066 (2005d:47007)
  • [Vi2005] -, Functional calculus under Kreiss type conditions, Math. Nachr. 278 (2005), no. 15, 1811-1822. MR 2182092 (2007b:47045)

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2010): 47A10

Retrieve articles in all journals with MSC (2010): 47A10


Additional Information

N. Nikolski
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia; University Bordeaux 1, France
Email: Nikolai.Nikolski@math.u-bordeaux1.fr

DOI: https://doi.org/10.1090/S1061-0022-2014-01295-2
Keywords: Power bounded, Kreiss Matrix Theorem, unconditional basis, Muckenhoupt condition
Received by editor(s): December 12, 2012
Published electronically: May 16, 2014
Dedicated: To Boris Mikhaĭlovich Makarov on his 80th anniversary — gratefully remembering unforgettable lessons in Analysis around 1960
Article copyright: © Copyright 2014 American Mathematical Society

American Mathematical Society