Proceedings of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6826(e) ISSN 0002-9939(p)

A Katznelson-Tzafriri type theorem in Hilbert spaces

Author(s): Zoltán Léka
Journal: Proc. Amer. Math. Soc. 137 (2009), 3763-3768.
MSC (2000): Primary 47A35, 46B08; Secondary 46M07, 47B99
Posted: May 27, 2009
MathSciNet review: 2529885
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

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

References:

1.: C.J.K. Batty, Asymptotic behaviour of semigroups of operators, in: Functional analysis and operator theory (J. Zemánek, ed.), Banach Center Publ., 30, Polish Acad. Sci., Warsaw, 1994, 35-52. MR 1285599 (95g:47058)
2.: C.J.K. Batty and S. Yeates, Extensions of semigroups of operators, J. Operator Theory, 46 (2001), 139-157. MR 1862183 (2002j:47067)
3.: C.J.K. Batty and Q.P. Vũ, Stability of strongly continuous representations of abelian semigroups, Math. Z., 209 (1992), 75-88. MR 1143215 (93d:47073)
4.: H. Bercovici, On the iterates of a completely nonunitary contraction, Topics in operator theory: Ernst D. Hellinger memorial volume, Operator Theory Adv. Appl., 48, Birkhäuser, Basel, 1990, 185-188. MR 1207397 (94g:47019)
5.: H. Bercovici, Commuting power bounded operators, Acta Sci. Math. (Szeged), 57 (1993), 55-64. MR 1243268 (94j:47008)
6.: R. Chill and Y. Tomilov, Stability of operator semigroups: ideas and results, Banach Center Publ., 75, Polish Acad. Sci., Warsaw, 2007, 71-109. MR 2336713 (2008m:47054)
7.: R.G. Douglas, On extending commutative semigroups of isometries, Bull. London Math. Soc., 1 (1969), 157-159. MR 0246153 (39:7458)
8.: J. Esterle, Quasimultipliers, representation of $H^\infty$ , and the closed ideal problem for commutative Banach algebras, in: Radical Banach algebras and automatic continuity, Lecture Notes in Math., 975, Springer, Berlin, 1983, 66-162. MR 697579 (85g:46067)
9.: J. Esterle, E. Strouse and F. Zouakia, Theorems of Katznelson-Tzafriri type for contractions, J. Funct. Anal., 94 (1990), 273-287. MR 1081645 (92c:47016)
10.: J. Esterle, E. Strouse and F. Zouakia, Stabilité asymptotique de certains semi-groupes d'opérateurs et idéaux primaires de $L^1(\mathbb{R}^+)$ , J. Operator Theory, 28 (1992), 203-227. MR 1273043 (95f:43001)
11.: S. Heinrich, Ultraproducts in Banach space theory, J. Reine Angew. Math., 313 (1980), 72-104. MR 552464 (82b:46013)
12.: T. Ito, On the commutative family of subnormal operators, J. Fac. Sci. Hokkaido Univ. Ser. I, 14 (1958), 1-15. MR 0107177 (21:5902)
13.: Y. Katznelson and L. Tzafriri, On power bounded operators, J. Funct. Anal., 68 (1986), 313-328. MR 859138 (88e:47006)
14.: L. Kérchy, Isometric asymptotes of power bounded operators, Indiana Univ. Math. J., 38 (1989), 173-188. MR 982576 (90f:47050)
15.: L. Kérchy and J. van Neerven, Polynomially bounded operators whose spectrum on the unit circle has measure zero, Acta Sci. Math. (Szeged), 63 (1997), 551-562. MR 1480498 (99a:47007)
16.: U. Krengel, Ergodic theorems, de Gruyter, Berlin, 1985. MR 797411 (87i:28001)
17.: H.S. Mustafayev, The Banach algebra generated by a contraction, Proc. Amer. Math. Soc., 134 (2006), 2677-2683. MR 2213747 (2006m:47013)
18.: 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), 19-34. MR 2301935 (2008b:47071)
19.: H. H. Schaefer, Banach lattices and positive operators, Grundl. Math. Wiss., 215, Springer-Verlag, New York- Heidelberg-Berlin, 1974. MR 0423039 (54:11023)
20.: Q. P. Vũ, A short proof of the Y. Katznelson's and L. Tzafriri's theorem, Proc. Amer. Math. Soc., 115 (1992), 1023-1024. MR 1087468 (92j:47012)
21.: Q. P. Vũ, Theorems of Katznelson-Tzafriri type for semigroups of operators, J. Funct. Anal., 103 (1992), 74-84. MR 1144683 (93e:47050)
22.: J. Zemánek, On the Gelfand-Hille theorems, in: Functional analysis and operator theory (J. Zemánek, ed.), Banach Center Publ., 30, Polish Acad. Sci., Warsaw, 1994, 369-385. MR 1285622 (95f:47009)

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|