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Proceedings of the American Mathematical Society

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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 $ 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.

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