Contractive commutants and invariant subspaces

Author:
R. L. Moore

Journal:
Proc. Amer. Math. Soc. **83** (1981), 747-750

MSC:
Primary 47A15

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]**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**

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

Article copyright:
© Copyright 1981
American Mathematical Society