## An Extension of Lomonosov’s Techniques to Non-compact Operators

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- by Aleksander Simonic
- Trans. Amer. Math. Soc.
**348**(1996), 975-995 - DOI: https://doi.org/10.1090/S0002-9947-96-01612-1
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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.## References

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## Bibliographic 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