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An Extension of Lomonosov's Techniques to non-compact Operators

Author: Aleksander Simonic
Journal: Trans. Amer. Math. Soc. 348 (1996), 975-995
MSC (1991): Primary 47A15; Secondary 46A32, 47D20
MathSciNet review: 1348869
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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.

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Additional Information

Aleksander Simonic
Affiliation: Department of Mathematics, Statistics & Computing Science, Dalhousie University, Halifax, Nova Scotia, B3H 3J5, Canada

Keywords: Linear operator, invariant subspace, transitive algebra
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
Article copyright: © Copyright 1996 American Mathematical Society