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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Extremal vectors and invariant subspaces


Authors: Shamim Ansari and Per Enflo
Journal: Trans. Amer. Math. Soc. 350 (1998), 539-558
MSC (1991): Primary 47A15
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 $K$ some weak limit of the sequence of minimal vectors is noncyclic for all operators commuting with $K$ and that for any normal operator $N$, the norm limit of the sequence of minimal vectors is noncyclic for all operators commuting with $N$. 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 $T$ belongs to a certain class ${\mathcal{C}}$ of operators, then the sequence of such vectors converges in norm, and that if $T$ belongs to a subclass of ${\mathcal{C}}$, then the norm limit is cyclic.


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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: http://dx.doi.org/10.1090/S0002-9947-98-01865-0
PII: S 0002-9947(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