• Communications of the Korean Mathematical Society
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• v.22 no.3
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• pp.383-389
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• 2007

In this paper we examine the hyperinvariant subspace problem for quasi-n-hyponormal operators. The main result on this problem is as follows. If T = N + K is such that N is a quasi-n-hyponormal operator whose spectrum contains an exposed arc and K belongs to the Schatten p-ideal then T has a non-trivial hyperinvariant subspace.

• Journal of the Chungcheong Mathematical Society
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• v.24 no.1
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• pp.19-34
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• 2011

A bounded linear operator T on a complex Banach space X is said to have property (I) provided that T has Bishop's property (${\beta}$) and there exists an integer p > 0 such that for a closed subset F of ${\mathbb{C}}$ ${X_T}(F)={E_T}(F)=\bigcap_{{\lambda}{\in}{\mathbb{C}}{\backslash}F}(T-{\lambda})^PX$ for all closed sets $F{\subseteq}{\mathbb{C}}$, where $X_T$(F) denote the analytic spectral subspace and $E_T$(F) denote the algebraic spectral subspace of T. Easy examples are provided by normal operators and hyponormal operators in Hilbert spaces, and more generally, generalized scalar operators and subscalar operators in Banach spaces. In this paper, we prove that if T has property (I), then the quasi-nilpotent part $H_0$(T) of T is given by $$KerT^P=\{x{\in}X:r_T(x)=0\}={\bigcap_{{\lambda}{\neq}0}(T-{\lambda})^PX$$ for all sufficiently large integers p, where ${r_T(x)}=lim\;sup_{n{\rightarrow}{\infty}}{\parallel}T^nx{\parallel}^{\frac{1}{n}}$. We also prove that if T has property (I) and the spectrum ${\sigma}$(T) is finite, then T is algebraic. Finally, we prove that if $T{\in}L$(X) has property (I) and has decomposition property (${\delta}$) then T has a non-trivial invariant closed linear subspace.