Lomonosov’s invariant subspace theorem for multivalued linear operators
HTML articles powered by AMS MathViewer
- by Peter Saveliev PDF
- Proc. Amer. Math. Soc. 131 (2003), 825-834 Request permission
Abstract:
The famous Lomonosov’s invariant subspace theorem states that if a continuous linear operator $T$ on an infinite-dimensional normed space $E$ “commutes” with a compact operator $K\neq 0,$ i.e., $TK=KT,$ then $T$ has a non-trivial closed invariant subspace. We generalize this theorem for multivalued linear operators. We also provide an application to single-valued linear operators.References
- Y. A. Abramovich, C. D. Aliprantis, and O. Burkinshaw, The invariant subspace problem: some recent advances, Rend. Istit. Mat. Univ. Trieste 29 (1998), no. suppl., 3–79 (1999). Workshop on Measure Theory and Real Analysis (Italian) (Grado, 1995). MR 1696022
- Ronald Cross, Multivalued linear operators, Monographs and Textbooks in Pure and Applied Mathematics, vol. 213, Marcel Dekker, Inc., New York, 1998. MR 1631548
- Ryszard Engelking, General topology, 2nd ed., Sigma Series in Pure Mathematics, vol. 6, Heldermann Verlag, Berlin, 1989. Translated from the Polish by the author. MR 1039321
- Lech Górniewicz, Topological fixed point theory of multivalued mappings, Mathematics and its Applications, vol. 495, Kluwer Academic Publishers, Dordrecht, 1999. MR 1748378, DOI 10.1007/978-94-015-9195-9
- V.I. Lomonosov, Invariant subspaces for operators commuting with compact operators, Functional Anal. Appl., 7 (1973), 213-214.
- Heydar Radjavi and Peter Rosenthal, Invariant subspaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 77, Springer-Verlag, New York-Heidelberg, 1973. MR 0367682
- Peter Saveliev, A Lefschetz-type coincidence theorem, Fund. Math. 162 (1999), no. 1, 65–89. MR 1734818
- Aleksander Simonič, A construction of Lomonosov functions and applications to the invariant subspace problem, Pacific J. Math. 175 (1996), no. 1, 257–270. MR 1419483
- Aleksander Simonič, An extension of Lomonosov’s techniques to non-compact operators, Trans. Amer. Math. Soc. 348 (1996), no. 3, 975–995. MR 1348869, DOI 10.1090/S0002-9947-96-01612-1
Additional Information
- Peter Saveliev
- Affiliation: Department of Mathematics, Allegheny College, Meadville, Pennsylvania 16335
- Address at time of publication: Department of Mathematics, Marshall University, Huntington, West Virginia 25755-2560
- Email: saveliev@member.ams.org
- Received by editor(s): September 19, 2000
- Received by editor(s) in revised form: October 14, 2001
- Published electronically: June 12, 2002
- Communicated by: Joseph A. Ball
- © Copyright 2002 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 131 (2003), 825-834
- MSC (2000): Primary 47A15, 47A06; Secondary 46A32, 54C60
- DOI: https://doi.org/10.1090/S0002-9939-02-06598-X
- MathSciNet review: 1937420