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
DOI: https://doi.org/10.1090/S0002-9939-1981-0630048-1
MathSciNet review: 630048
Full-text PDF Free Access

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] S. Brown, Connections between an operator and a compact operator that yield hyperinvariant subspaces, J. Operator Theory 1 (1979), 117-122. MR 526293 (80h:47005)
  • [2] C. K. Fong, A note on common invariant subspaces (to appear). MR 658617 (84g:47003)
  • [3] P. R. Halmos, A Hilbert space problem book, Van Nostrand, Princeton, N.J., 1967. MR 0208368 (34:8178)
  • [4] H. W. Kim, R. L. Moore and C. M. Pearcy, A variation of Lomonosov's theorem, J. Operator Theory 2 (1979), 131-140. MR 553868 (81b:47007)
  • [5] V. I. Lomonosov, On invariant subspaces of families of operators commuting with a completely continuous operator, Funkcional. Anal. i Priložen. 7 (1973), 55-56. MR 0420305 (54:8319)
  • [6] Carl Pearcy and Allen L. Shields, A survey of the Lomonosov technique in the theory of invariant subspaces, Topics in Operator Theory, Math. Surveys, no. 13, Amer. Math. Soc., Providence, R. I., 1974, pp. 219-229. MR 0355639 (50:8113)

Similar Articles

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

Retrieve articles in all journals with MSC: 47A15


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1981-0630048-1
Article copyright: © Copyright 1981 American Mathematical Society

American Mathematical Society