Invariant subspaces for Banach space operators with an annular spectral set

Yavuz, Onur

doi:10.1090/S0002-9947-07-04324-3

Invariant subspaces for Banach space operators with an annular spectral set
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Trans. Amer. Math. Soc. 360 (2008), 2661-2680 Request permission

Abstract:

Consider an annulus $\Omega =\{z\in \mathbb {C}:r_ {0}<|z|<1\}$ for some $0<r_{0}<1$, and let $T$ be a bounded invertible linear operator on a Banach space $X$ whose spectrum contains $\partial \Omega$. Assume there exists a constant $K>0$ such that $\|p(T)\|~\leq ~ K \sup \{|p(\lambda )|:|\lambda |\leq 1\}$ and $\|p(r_0T^{-1})\|\leq K \sup \{|p(\lambda )|:|\lambda |\leq 1\}$ for all polynomials $p$. Then there exists a nontrivial common invariant subspace for $T^{*}$ and ${T^{*}}^{-1}$.

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