Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Contractive commutants and invariant subspaces

Author: R. L. Moore
Journal: Proc. Amer. Math. Soc. 83 (1981), 747-750
MSC: Primary 47A15
MathSciNet review: 630048
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ T$ be a bounded operator on a Banach space $ \mathfrak{X}$ and let $ K$ be a nonzero compact operator. In [1] and [4] it is shown that if $ \lambda $ is a complex number and if $ TK=\lambda KT $, then $ T$ has a hyperinvariant subspace. In [1], S. Brown goes on to show that if $ \mathfrak{X}$ is reflexive and if $ TK = \lambda KT$ and $ TB = \mu BT$ for some $ \lambda $, $ \mu $ with $ \left\vert \lambda \right\vert \ne 1$ and $ (1 - \left\vert \mu \right\vert)/(1 - \left\vert \lambda \right\vert) \geqslant 0$, then $ B$ has an invariant subspace. Below we extend both these results by showing that the entire class of operators satisfying the above conditions on $ B$ has an invariant subspace.

References [Enhancements On Off] (What's this?)

  • [1] Scott Brown, Connections between an operator and a compact operator that yield hyperinvariant subspaces, J. Operator Theory 1 (1979), no. 1, 117–121. MR 526293
  • [2] C. K. Fong, A note on common invariant subspaces, J. Operator Theory 7 (1982), no. 2, 335–339. MR 658617
  • [3] Paul R. Halmos, A Hilbert space problem book, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR 0208368
  • [4] H. W. Kim, R. Moore, and C. M. Pearcy, A variation of Lomonosov’s theorem, J. Operator Theory 2 (1979), no. 1, 131–140. MR 553868
  • [5] V. I. Lomonosov, Invariant subspaces of the family of operators that commute with a completely continuous operator, Funkcional. Anal. i Priložen. 7 (1973), no. 3, 55–56 (Russian). MR 0420305
  • [6] Carl Pearcy and Allen L. Shields, A survey of the Lomonosov technique in the theory of invariant subspaces, Topics in operator theory, Amer. Math. Soc., Providence, R.I., 1974, pp. 219–229. Math. Surveys, No. 13. MR 0355639

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 47A15

Retrieve articles in all journals with MSC: 47A15

Additional Information

Article copyright: © Copyright 1981 American Mathematical Society

American Mathematical Society