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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Extremal vectors and invariant subspaces
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by Shamim Ansari and Per Enflo PDF
Trans. Amer. Math. Soc. 350 (1998), 539-558 Request permission

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.
References
  • Bernard Beauzamy, Introduction to operator theory and invariant subspaces, North-Holland Mathematical Library, vol. 42, North-Holland Publishing Co., Amsterdam, 1988. MR 967989
  • M. Ĭ. Kadec′, Spaces isomorphic to a locally uniformly convex space, Izv. Vysš. Učebn. Zaved. Matematika 1959 (1959), no. 6 (13), 51–57 (Russian). MR 0126692
  • P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
  • Heydar Radjavi and Peter Rosenthal, Invariant subspaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 77, Springer-Verlag, New York-Heidelberg, 1973. MR 0367682, DOI 10.1007/978-3-642-65574-6
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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
  • Received by editor(s): October 16, 1995
  • Additional Notes: Partially supported by NSF grant number 441003
  • © Copyright 1998 American Mathematical Society
  • 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