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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Derivations and (hyper)invariant subspaces of a bounded operator
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by Shuang Zhang PDF
Proc. Amer. Math. Soc. 102 (1988), 261-267 Request permission

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.
References
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Additional Information
  • © Copyright 1988 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 102 (1988), 261-267
  • MSC: Primary 47A15,; Secondary 47B47
  • DOI: https://doi.org/10.1090/S0002-9939-1988-0920983-5
  • MathSciNet review: 920983