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

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Abstract: Let be a bounded operator on a Banach space and let be a nonzero compact operator. In [**1**] and [**4**] it is shown that if is a complex number and if , then has a hyperinvariant subspace. In [**1**], S. Brown goes on to show that if is reflexive and if and for some , with and , then has an invariant subspace. Below we extend both these results by showing that the entire class of operators satisfying the above conditions on has an invariant subspace.

**[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)**

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DOI:
https://doi.org/10.1090/S0002-9939-1981-0630048-1

Article copyright:
© Copyright 1981
American Mathematical Society