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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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An Extension of Lomonosov’s Techniques to Non-compact Operators
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by Aleksander Simonic PDF
Trans. Amer. Math. Soc. 348 (1996), 975-995 Request permission

Abstract:

The aim of this work is to generalize Lomonosov’s techniques in order to apply them to a wider class of not necessarily compact operators. We start by establishing a connection between the existence of invariant subspaces and density of what we define as the associated Lomonosov space in a certain function space. On a Hilbert space, approximation with Lomonosov functions results in an extended version of Burnside’s Theorem. An application of this theorem to the algebra generated by an essentially self-adjoint operator $A$ yields the existence of vector states on the space of all polynomials restricted to the essential spectrum of $A$. Finally, the invariant subspace problem for compact perturbations of self-adjoint operators acting on a real Hilbert space is translated into an extreme problem and the solution is obtained upon differentiating certain real-valued functions at their extreme.
References
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Additional Information
  • Aleksander Simonic
  • Affiliation: Department of Mathematics, Statistics & Computing Science, Dalhousie University, Halifax, Nova Scotia, B3H 3J5, Canada
  • Email: alex@cs.dal.ca
  • Received by editor(s): February 15, 1995
  • Additional Notes: This work was completed with the support of an Izaak Walton Killam Memorial Scholarship.
    The author was also supported in part by the Research Council of Slovenia.
  • Communicated by: Daniel J. Rudolph
  • © Copyright 1996 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 348 (1996), 975-995
  • MSC (1991): Primary 47A15; Secondary 46A32, 47D20
  • DOI: https://doi.org/10.1090/S0002-9947-96-01612-1
  • MathSciNet review: 1348869