Operators in the commutant of a reductive algebra

Moore, Robert L.

doi:10.1090/S0002-9939-1977-0454730-3

Operators in the commutant of a reductive algebra
HTML articles powered by AMS MathViewer

by Robert L. Moore

Proc. Amer. Math. Soc. 66 (1977), 99-104

DOI: https://doi.org/10.1090/S0002-9939-1977-0454730-3

PDF | Request permission

Abstract:

Let $\mathcal {A}$ be a reductive algebra. It is shown that there is a subspace $\mathcal {M}$ that reduces $\mathcal {A}$ and such that the commutant of $\mathcal {A}|\mathcal {M}$ is selfadjoint and the commutant of $\mathcal {A}|{\mathcal {M}^ \bot }$ consists of hyporeductive operators. It is then shown that under a variety of conditions, if an operator T is in $\mathcal {A}’$, then ${T^ \ast }$ is in $\mathcal {A}’$.

References

On operators with reducing hyperinvariant subspaces

Che Kao Fong, On commutants of reductive algebras, Proc. Amer. Math. Soc. 63 (1977), no. 1, 111–114. MR 440416, DOI 10.1090/S0002-9939-1977-0440416-8
Paul R. Halmos, A Hilbert space problem book, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR 0208368
T. B. Hoover, Operator algebras with reducing invariant subspaces, Pacific J. Math. 44 (1973), 173–179. MR 315477
Robert L. Moore, Reductive operators that commute with a compact operator, Michigan Math. J. 22 (1975), no. 3, 229–233 (1976). MR 394246
Robert L. Moore, Hyperinvariant subspaces of reductive operators, Proc. Amer. Math. Soc. 63 (1977), no. 1, 91–94. MR 435888, DOI 10.1090/S0002-9939-1977-0435888-9
Carl Pearcy, J. R. Ringrose, and Norberto Salinas, Remarks on the invariant-subspace problem, Michigan Math. J. 21 (1974), 163–166. MR 343062
Heydar Radjavi and Peter Rosenthal, Invariant subspaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 77, Springer-Verlag, New York-Heidelberg, 1973. MR 0367682
Peter Rosenthal, On commutants of reductive operator algebras, Duke Math. J. 41 (1974), 829–834. MR 358395