Abstract
The Katznelson-Tzafriri Theorem states that, given a power-bounded operator T, ǁT n(I − T)ǁ → 0 as n → ∞ if and only if the spectrum σ(T) of T intersects the unit circle T in at most the point 1. This paper investigates the rate at which decay takes place when σ(T) ∩ T = {1}. The results obtained lead, in particular, to both upper and lower bounds on this rate of decay in terms of the growth of the resolvent operator R(eiθ, T) as θ → 0. In the special case of polynomial resolvent growth, these bounds are then shown to be optimal for general Banach spaces but not in the Hilbert space case.
Similar content being viewed by others
References
G. R. Allan and T. J. Ransford, Power-dominated elements in a Banach algebra, Studia Math. 94 (1989), 63–79.
W. Arendt, C. J. K. Batty, M. Hieber, and F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems, second edition, Birkhäuser/Springer Basel AG, Basel, 2011.
C. Arhancet and C. Le Merdy, Dilation of Ritt operators on Lp-spaces, Israel J. Math. 201 (2014), 373–414.
C. Badea and Y. I. Lyubich, Geometric, spectral and asymptotic properties of averaged products of projections in Banach spaces, Studia Math. 201 (2010), 21–35.
A. Bátkai, K.-J. Engel, J. Prüss, and R. Schnaubelt, Polynomial stability of operator semigroups, Math. Nachr. 279 (2006), 1425–1440.
C. J. K. Batty, Asymptotic behaviour of semigroups of operators, in Functional Analysis and Operator Theory, Polish Acad. Sci., Warsaw, 1994, pp. 35–52.
C. J. K. Batty, R. Chill, and Y. Tomilov, Fine scales of decay of operator semigroups, J. Eur. Math. Soc. (JEMS) 18 (2016), 853–924.
C. J. K. Batty and T. Duyckaerts, Non-uniform stability for bounded semigroups on Banach spaces, J. Evol. Equ. 8 (2008), 765–780.
S. Blunck, Analyticity and discrete maximal regularity on Lp-spaces, J. Funct. Anal. 183 (2001), 211–230.
S. Blunck, Maximal regularity of discrete and continuous time evolution equations, Studia Math. 146 (2001), 157–176.
A. A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Ann. 347 (2010), 455–478.
R. Chill and Y. Tomilov, Stability of operator semigroups: ideas and results, in Perspectives in Operator Theory, Polish Acad. Sci., Warsaw, 2007, pp. 71–109.
P. Diaconis, G. Lebeau, and L. Michel, Geometric analysis for the Metropolis algorithm on Lipschitz domains, Invent. Math. 185 (2011), 239–281.
N. Dungey, Time regularity for random walks on locally compact groups, Probab. Theory Related Fields 137 (2007), 429–442.
N. Dungey, On time regularity and related conditions for power-bounded operators, Proc. Lond. Math. Soc. (3) 97 (2008), 97–116.
N. Dungey, Time regularity for aperiodic or irreducible random walks on groups, Hokkaido Math. J. 37 (2008), 19–40.
N. Dungey, Subordinated discrete semigroups of operators, Trans. Amer. Math. Soc. 363 (2011), 1721–1741.
O. El-Fallah and T. Ransford, Extremal growth and powers of operators satisfying resolvent conditions of Kreiss-Ritt type, J. Funct. Anal. 196 (2002), 135–154, 2002.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, Cambridge University Press, Cambridge, 2009.
A. M. Gomilko, Cayley transform of the generator of a uniformly bounded C0-semigroup of operators Ukrainian Math. J. 56 (2004), 1212–1226.
G. Grimmett and D. Welsh, Probability: An Introduction, Oxford University Press, 1986.
M. Haase, A functional calculus description of real interpolation spaces for sectorial operators, Studia Math. 171 (2005), 177–195.
M. Haase and Y. Tomilov, Domain characterizations of certain functions of power-bounded operators, Studia Math. 196 (2010), 265–288.
N. Kalton, S. Montgomery-Smith, K. Olieszkiewicz, and Y. Tomilov, Power-bounded operators and related norm estimates, J. Lond. Math. Soc. (2) 70 (2004), 463–478.
Y. Katznelson and L. Tzafriri, On power bounded operators, J. Funct. Anal. 68 (1986), 313–328.
P. Koosis, The Logarithmic Integral, Volume 1, Cambridge University Press, Cambridge, 1988.
F. Lancien and C. Le Merdy, On functional calculus properties of Ritt operators, Proc. Roy. Soc. Edinburgh Sect. A 145 (2015), 1239–1250.
Y. Latushkin and R. Shvydkoy, Hyperbolicity of semigroups and Fourier multipliers, in Systems, Approximations, Singular Integral Operators, and Related Topics, Birkhäuser, Basel, 2001, pp. 341–363.
Z. Léka, A Katznelson-Tzafriri type theorem in Hilbert spaces, Proc. Amer. Math. Soc. 137 (2009), 3763–3768.
Z. Léka, Time regularity and functions of the Volterra operator, Studia Math. 20 (2014), 1–14.
C. Le Merdy. H8 functional calculus and square function estimates for Ritt operators, Rev. Mat. Iberoam. 30 (2014), 1149–1190.
C. Le Merdy and Q. Xu, Maximal theorems and square functions for analytic operators on Lpspaces, J. Lond. Math. Soc. (2) 86 (2012), 343–365.
Y. Lyubich, Spectral localization, power boundedness and invariant subspaces under Ritt’s type condition, Studia Math. 134 (1999), 153–167.
M. M. Martínez, Decay estimates of functions through singular extensions of vector-valued Laplace transforms, J. Math. Anal. Appl. 375 (2011), 196–206.
B. Nagy and J. Zemánek, A resolvent condition implying power boundedness, Studia Math. 134 (1999), 143–151.
O. Nevanlinna, Convergence of Iterations for Linear Equations, Birkhäuser, Basel, 1993.
O. Nevanlinna, On the growth of the resolvent operators for power bounded operators, in Linear Operators, Polish Acad. Sci., Warsaw, 1997, pp. 247–264.
O. Nevanlinna, Resolvent conditions and powers of operators, Studia Math. 145 (2001), 113–134.
D. Ornstein and L. Sucheston, An operator theorem on L1 convergence to zero with applications to Markov kernels, Ann. Math. Statist. 41 (1970), 1631–1639.
D. Seifert, Some improvements of the Katznelson-Tzafriri theorem on Hilbert space, Proc. Amer. Math. Soc. 143 (2015), 3827–3838.
D. Seifert, A quantified Tauberian theorem for sequences, Studia Math. 227 (2015), 183–192.
P. Vitse, Functional calculus under the Tadmor-Ritt condition, and free interpolation by polynomials of a given degree, J. Funct. Anal. 210 (2004), 43–72.
M. Zarrabi, Some results of Katznelson-Tzafriri type, J. Math. Anal. Appl. 397 (2013), 109–118.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Seifert, D. Rates of decay in the classical Katznelson-Tzafriri theorem. JAMA 130, 329–354 (2016). https://doi.org/10.1007/s11854-016-0039-3
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11854-016-0039-3