Asymptotic behavior of polynomially bounded operators

Heybetkulu S. Mustafayev

doi:10.1016/j.crma.2010.04.003

Mathematical Analysis

Asymptotic behavior of polynomially bounded operators

Presented by: Gilles Pisier

Heybetkulu S. Mustafayev ¹

¹Yuzuncu Yıl University, Faculty of Arts and Sciences, Department of Mathematics, 65080, Van, Turkey

Comptes Rendus. Mathématique, Volume 348 (2010) no. 9-10, pp. 517-520

Abstract (Original language)
Résumé

Let T be a polynomially bounded operator on a complex Banach space and let $A_{T}$ be the smallest uniformly closed (Banach) algebra that contains T and the identity operator. It is shown that for every $S \in A_{T}$ ,

\lim_{n \to \infty} ‖ T^{n} S ‖ = \sup_{ξ \in σ_{u} (T)} | \hat{S} (ξ) |,

where

\hat{S}

is the Gelfand transform of S and

σ_{u} (T) : = σ (T) \cap Γ

is the unitary spectrum of T;

Γ = {z \in C : | z | = 1}

.

Soit T un opérateur polynomialement borné sur un espace de Banach et soit $A_{T}$ la plus petite algèbre de Banach uniformement fermé contenant T et l'identité. Il est montré dans cet article que pour tout $S \in A_{T}$ ,

\lim_{n \to \infty} ‖ T^{n} S ‖ = \sup_{ξ \in σ_{u} (T)} | \hat{S} (ξ) |,

où

\hat{S}

est la transformée de Gelfand et

σ_{u} (T) : = σ (T) \cap Γ

est la spectre unitaire de T ;

Γ : = {z \in C : | z | = 1}

.

Received: 2010-01-26
Accepted: 2010-04-08
Published online: 2010-04-24

DOI: 10.1016/j.crma.2010.04.003

Author's affiliations:

Heybetkulu S. Mustafayev ¹

¹ Yuzuncu Yıl University, Faculty of Arts and Sciences, Department of Mathematics, 65080, Van, Turkey

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     title = {Asymptotic behavior of polynomially bounded operators},
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     pages = {517--520},
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     doi = {10.1016/j.crma.2010.04.003},
     language = {en},
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Heybetkulu S. Mustafayev. Asymptotic behavior of polynomially bounded operators. Comptes Rendus. Mathématique, Volume 348 (2010) no. 9-10, pp. 517-520. doi: 10.1016/j.crma.2010.04.003

References
Cited by

[1] J. Esterle; E. Strouse; F. Zouakia Theorems of Katznelson–Tzafriri type for contractions, J. Funct. Anal., Volume 94 (1990), pp. 273-287

[2] Y. Katznelson; L. Tzafriri On power bounded operators, J. Funct. Anal., Volume 68 (1986), pp. 313-328

[3] L. Kérchy; J. van Neerven Polynomially bounded operators whose spectrum on the unit circle has measure zero, Acta Sci. Math. (Szeged), Volume 63 (1997), pp. 551-562

[4] R. Larsen Banach Algebras, Marcel-Dekker, Inc., New York, 1973

[5] H. Mustafayev Dissipative operators on Banach spaces, J. Funct. Anal., Volume 248 (2007), pp. 428-447

[6] N.K. Nikolski Treatise on the Shift Operator, Nauka, Moscow, 1980 (in Russian)

[7] V.Q. Phóng Theorems of Katznelson–Tzafriri type for semigroups of operators, J. Funct. Anal., Volume 94 (1990), pp. 273-287

[8] G. Pisier A polynomially bounded operator on Hilbert space which is not similar to a contraction, J. Amer. Math. Soc., Volume 10 (1997), pp. 351-369

[9] B. Sz.-Nagy; C. Foias Harmonic Analysis of Operators on Hilbert Space, North-Holland, Amsterdam, 1970

[10] M. Zarrabi On polynomially bounded operators acting on a Banach space, J. Funct. Anal., Volume 225 (2005), pp. 147-166

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