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Proceedings of the American Mathematical Society

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



Derivations and (hyper)invariant subspaces of a bounded operator

Author: Shuang Zhang
Journal: Proc. Amer. Math. Soc. 102 (1988), 261-267
MSC: Primary 47A15,; Secondary 47B47
MathSciNet review: 920983
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Abstract: Let $ X$ be a complex Banach space and $ \mathcal{L}\left( X \right)$ the set of bounded linear operators on $ X$. For $ T \in \mathcal{L}\left( X \right)$, a derivation $ {\Delta _T}$ is defined by $ {\Delta _T}A = TA - AT$ for $ A \in \mathcal{L}\left( X \right)$. By induction, $ \Delta _T^m = {\Delta _T} \circ \Delta _T^{m - 1}$ is defined for each integer $ m \geq 2$. We call the kernel of $ \Delta _T^m$ the $ m$-commutant of $ T$. For a polynomially compact operator $ T$, we consider the (hyper)invariant subspace problem for operators in the $ m$-commutant of $ T$ for $ m \geq 1$. It is easily seen that the $ m$-commutant $ m > 1$ of $ T$ could be much larger than $ {\operatorname{Ker}}({\Delta _T})$. So our idea is a variation of Lomonosov's theorem in [6]. We start with several identities on derivations, and then prove our results on the existence of (hyper)invariant subspaces. Theorem 2 in [5] is generalized.

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Keywords: Invariant subspaces, commutators, derivations
Article copyright: © Copyright 1988 American Mathematical Society