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Convergence properties of minimal vectors for normal operators and weighted shifts
Author(s):
Isabelle
Chalendar;
Jonathan
R.
Partington
Journal:
Proc. Amer. Math. Soc.
133
(2005),
501-510.
MSC (2000):
Primary 41A29, 47A15, 47A16, 47B20
Posted:
September 8, 2004
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Additional information
Abstract:
We study the behaviour of the sequence of minimal vectors corresponding to certain classes of operators on reflexive  spaces, including multiplication operators and bilateral weighted shifts. The results proved are based on explicit formulae for the minimal vectors, and provide extensions of results due to Ansari and Enflo, and also Wiesner. In many cases the convergence of sequences associated with the minimal vectors leads to the construction of hyperinvariant subspaces for cyclic operators.
References:
-
- 1.
- G. Androulakis, A note on the method of minimal vectors. Trends in Banach spaces and operator theory, Contemp. Math., 321:29-36, 2003.
- 2.
- S. Ansari and P. Enflo. Extremal vectors and invariant subspaces. Trans. Amer. Math. Soc., 350:539-558, 1998. MR 98d:47019
- 3.
- I. Chalendar and J. R. Partington. Approximation problems and representations of Hardy spaces in circular domains. Studia Math., 136(3):255-269, 1999. MR 2000i:30063
- 4.
- I. Chalendar and J. R. Partington. Constrained approximation and invariant subspaces. J. Math. Anal. Appl., 280:176-187, 2003. MR 2004d:41027
- 5.
- I. Chalendar, J. R. Partington, and M. Smith. Approximation in reflexive Banach spaces and applications to the invariant subspace problem. Proc. Amer. Math. Soc, 2003.
- 6.
- J. B. Conway. The theory of subnormal operators, volume 36 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 1991. MR 92h:47026
- 7.
- P. Enflo. Extremal vectors for a class of linear operators. In Functional analysis and economic theory (Samos, 1996), pages 61-64. Springer, Berlin, 1998. MR 2000k:47020
- 8.
- P. Enflo and T. Hõim. Some results on extremal vectors and invariant subspaces. Proc. Amer. Math. Soc., 131(2):379-387, 2003. MR 2003i:47007
- 9.
- R. Frankfurt. Subnormal weighted shifts and related function spaces. J. Math. Anal. Appl., 52(3):471-489, 1975. MR 58:2407
- 10.
- D. A. Herrero. Eigenvectors and cyclic vectors for bilateral weighted shifts. Rev. Un. Mat. Argentina, 26:24-41, 1972/73. MR 49:1170
- 11.
- D. Hitt. Invariant subspaces of
 of an annulus. Pacific J. Math., 134(1):101-120, 1988. MR 90a:46059 - 12.
- H. Radjavi and P. Rosenthal. Invariant subspaces. Springer-Verlag, New York, 1973. MR 51:3924
- 13.
- D. Sarason. Nearly invariant subspaces of the backward shift. In Contributions to operator theory and its applications (Mesa, AZ, 1987), volume 35 of Oper. Theory Adv. Appl., pages 481-493. Birkhäuser, Basel, 1988. MR 90m:47012
- 14.
- A. Spalsbury. Vectors of minimal norm. Proc. Amer. Math. Soc., 350:2737-2745, 1998. MR 98k:47009
- 15.
- H. A. Taha. Operations research. Macmillan Co., New York, 1982. MR 88f:90057a
- 16.
- V. G. Troitsky. Minimal vectors in arbitrary Banach spaces. Proc. Amer. Math. Soc., to appear.
- 17.
- E. Wiesner. Backward minimal points for bounded linear operators on finite-dimensional vector spaces. Linear Algebra Appl., 338:251-259, 2001. MR 2002g:47033
- 18.
- D. V. Yakubovich. Invariant subspaces of the operator of multiplication by
 in the space  in a multiply connected domain. J. Soviet Math., 61(2):2046-2056, 1992. MR 91c:47061
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Additional Information:
Isabelle
Chalendar
Affiliation:
Institut Girard Desargues, UFR de Mathématiques, Université Claude Bernard Lyon~1, 69622 Villeurbanne Cedex, France
Email:
chalenda@igd.univ-lyon1.fr
Jonathan
R.
Partington
Affiliation:
School of Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom
Email:
J.R.Partington@leeds.ac.uk
DOI:
10.1090/S0002-9939-04-07595-1
PII:
S 0002-9939(04)07595-1
Keywords:
Minimal vectors,
hyperinvariant subspaces,
multiplication operators,
weighted shifts,
hyponormal operators
Received by editor(s):
July 23, 2003
Received by editor(s) in revised form:
October 16, 2003
Posted:
September 8, 2004
Communicated by:
N. Tomczak-Jaegermann
Copyright of article:
Copyright
2004,
American Mathematical Society
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