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