A Katznelson-Tzafriri type theorem in Hilbert spaces

Léka, Zoltán

doi:10.1090/S0002-9939-09-09939-0

A Katznelson-Tzafriri type theorem in Hilbert spaces
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by Zoltán Léka

Proc. Amer. Math. Soc. 137 (2009), 3763-3768

DOI: https://doi.org/10.1090/S0002-9939-09-09939-0

Published electronically: May 27, 2009

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Abstract:

Our aim is to characterize, via an ergodic condition, the norm convergence $\lim _{n \rightarrow \infty } \|T^nQ\| = 0$ when $T$ is a power-bounded operator on a Hilbert space and $Q$ commutes with $T.$ We shall also prove that if $f \in A^+(\mathbb {T})$ and $Q = f(T),$ the given condition is equivalent to the vanishing of $f$ on the peripheral spectrum of $T.$

References

C. J. K. Batty, Asymptotic behaviour of semigroups of operators, Functional analysis and operator theory (Warsaw, 1992) Banach Center Publ., vol. 30, Polish Acad. Sci. Inst. Math., Warsaw, 1994, pp. 35–52. MR 1285599
Charles J. K. Batty and Stephen B. Yeates, Extensions of semigroups of operators, J. Operator Theory 46 (2001), no. 1, 139–157. MR 1862183
Charles J. K. Batty and Vũ Quốc Phóng, Stability of strongly continuous representations of abelian semigroups, Math. Z. 209 (1992), no. 1, 75–88. MR 1143215, DOI 10.1007/BF02570822
Hari Bercovici, On the iterates of a completely nonunitary contraction, Topics in operator theory: Ernst D. Hellinger memorial volume, Oper. Theory Adv. Appl., vol. 48, Birkhäuser, Basel, 1990, pp. 185–188. MR 1207397
Hari Bercovici, Commuting power-bounded operators, Acta Sci. Math. (Szeged) 57 (1993), no. 1-4, 55–64. MR 1243268
Ralph Chill and Yuri Tomilov, Stability of operator semigroups: ideas and results, Perspectives in operator theory, Banach Center Publ., vol. 75, Polish Acad. Sci. Inst. Math., Warsaw, 2007, pp. 71–109. MR 2336713, DOI 10.4064/bc75-0-6
R. G. Douglas, On extending commutative semigroups of isometries, Bull. London Math. Soc. 1 (1969), 157–159. MR 246153, DOI 10.1112/blms/1.2.157
J. Esterle, Quasimultipliers, representations of $H^{\infty }$, and the closed ideal problem for commutative Banach algebras, Radical Banach algebras and automatic continuity (Long Beach, Calif., 1981) Lecture Notes in Math., vol. 975, Springer, Berlin, 1983, pp. 66–162. MR 697579, DOI 10.1007/BFb0064548
J. Esterle, E. Strouse, and F. Zouakia, Theorems of Katznelson-Tzafriri type for contractions, J. Funct. Anal. 94 (1990), no. 2, 273–287. MR 1081645, DOI 10.1016/0022-1236(90)90014-C
J. Esterle, E. Strouse, and F. Zouakia, Stabilité asymptotique de certains semi-groupes d’opérateurs et idéaux primaires de $L^1(\textbf {R}^+)$, J. Operator Theory 28 (1992), no. 2, 203–227 (French). MR 1273043
Stefan Heinrich, Ultraproducts in Banach space theory, J. Reine Angew. Math. 313 (1980), 72–104. MR 552464, DOI 10.1515/crll.1980.313.72
Takasi Itô, On the commutative family of subnormal operators, J. Fac. Sci. Hokkaido Univ. Ser. I 14 (1958), 1–15. MR 0107177
Y. Katznelson and L. Tzafriri, On power bounded operators, J. Funct. Anal. 68 (1986), no. 3, 313–328. MR 859138, DOI 10.1016/0022-1236(86)90101-1
László Kérchy, Isometric asymptotes of power bounded operators, Indiana Univ. Math. J. 38 (1989), no. 1, 173–188. MR 982576, DOI 10.1512/iumj.1989.38.38008
László Kérchy and Jan van Neerven, Polynomially bounded operators whose spectrum on the unit circle has measure zero, Acta Sci. Math. (Szeged) 63 (1997), no. 3-4, 551–562. MR 1480498
Ulrich Krengel, Ergodic theorems, De Gruyter Studies in Mathematics, vol. 6, Walter de Gruyter & Co., Berlin, 1985. With a supplement by Antoine Brunel. MR 797411, DOI 10.1515/9783110844641
H. S. Mustafayev, The Banach algebra generated by a contraction, Proc. Amer. Math. Soc. 134 (2006), no. 9, 2677–2683. MR 2213747, DOI 10.1090/S0002-9939-06-08302-X
H. S. Mustafayev, The behavior of the radical of the algebras generated by a semigroup of operators on Hilbert space, J. Operator Theory 57 (2007), no. 1, 19–34. MR 2301935
Helmut H. Schaefer, Banach lattices and positive operators, Die Grundlehren der mathematischen Wissenschaften, Band 215, Springer-Verlag, New York-Heidelberg, 1974. MR 0423039
Vũ Quốc Phóng, A short proof of the Y. Katznelson’s and L. Tzafriri’s theorem, Proc. Amer. Math. Soc. 115 (1992), no. 4, 1023–1024. MR 1087468, DOI 10.1090/S0002-9939-1992-1087468-0
Vũ Quốc Phóng, Theorems of Katznelson-Tzafriri type for semigroups of operators, J. Funct. Anal. 103 (1992), no. 1, 74–84. MR 1144683, DOI 10.1016/0022-1236(92)90135-6
Jaroslav Zemánek, On the Gel′fand-Hille theorems, Functional analysis and operator theory (Warsaw, 1992) Banach Center Publ., vol. 30, Polish Acad. Sci. Inst. Math., Warsaw, 1994, pp. 369–385. MR 1285622