• 대한수학회논문집
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• 제12권3호
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• pp.561-569
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• 1997

In this paper we shall characterize a finite triangular operator matrix with M-hyponormal operators on main diagonal. This shows in particualr that such an operator is subscalar operator. As a corollary, we get that every algebraic operator is subscalar.

• 대한수학회보
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• 제42권3호
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• pp.571-577
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• 2005

In this paper we prove that if T is a k-th root of a p­hyponormal operator when T is compact or T$^{n}$ is normal for some integer n > k, then T is (generalized) scalar, and that if T is a k-th root of a semi-hyponormal operator and have the property $\sigma$(T) is contained in an angle < 2$\pi$/k with vertex in the origin, then T is subscalar.

• 충청수학회지
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• 제24권1호
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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.