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

DOI:
https://doi.org/10.1090/S0002-9947-96-01612-1

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

Email:
alex@cs.dal.ca

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