Extremal vectors and invariant subspaces

Authors:
Shamim Ansari and Per Enflo

Journal:
Trans. Amer. Math. Soc. **350** (1998), 539-558

MSC (1991):
Primary 47A15

DOI:
https://doi.org/10.1090/S0002-9947-98-01865-0

MathSciNet review:
1407476

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Abstract: For a bounded linear operator on Hilbert space we define a sequence of so-called minimal vectors in connection with invariant subspaces and show that this presents a new approach to invariant subspaces. In particular, we show that for any compact operator some weak limit of the sequence of minimal vectors is noncyclic for all operators commuting with and that for any normal operator , the norm limit of the sequence of minimal vectors is noncyclic for all operators commuting with . Thus, we give a new and more constructive proof of existence of invariant subspaces. The sequence of minimal vectors does not seem to converge in norm for an arbitrary bounded linear operator. We will prove that if belongs to a certain class of operators, then the sequence of such vectors converges in norm, and that if belongs to a subclass of , then the norm limit is cyclic.

**1.**B. Beauzamy,*Introduction to Operator Theory and Invariant Subspaces*, North-Holland (1988). MR**90d:47001****2.**M. I. Kadets,*On spaces isomorphic to locally uniformly rotund spaces*, Izv. Vys\v{s}. U\v{c}ebn. Zaved. Mathematika**1959**, no. 6, 51-57. (Russian) MR**23:A3987****3.**S. Mazur,*On the generalized limit of bounded sequences*, Colloquium Mathematicum, 2, (1949-1951), 173-175. MR**14:159k****4.**H. Radjavi and P. Rosenthal,*Invariant Subspaces*, New York: Springer-Verlag, 1973. MR**51:3924**

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Additional Information

**Shamim Ansari**

Affiliation:
Department of Mathematics, Kent State University, Kent, Ohio 44242

Address at time of publication:
Department of Mathematics & Statistics, Drawer MA, Mississippi State University, Mississippi State, Mississippi 39762

**Per Enflo**

Affiliation:
Department of Mathematics, Kent State University, Kent, Ohio 44242

DOI:
https://doi.org/10.1090/S0002-9947-98-01865-0

Received by editor(s):
October 16, 1995

Additional Notes:
Partially supported by NSF grant number 441003

Article copyright:
© Copyright 1998
American Mathematical Society