• Communications of the Korean Mathematical Society
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• v.12 no.3
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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.

• Bulletin of the Korean Mathematical Society
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• v.48 no.6
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• pp.1129-1135
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• 2011

In this note, we prove that every subscalar operator with finite spectrum is algebraic. In particular, a quasi-nilpotent subscala operator is nilpotent. We also prove that every subscalar operator with property (${\delta}$) on a Banach space of dimension greater than 1 has a nontrivial invariant closed linear subspace.

• Bulletin of the Korean Mathematical Society
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• v.35 no.4
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• pp.701-707
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• 1998

An operator $T{\in} {{\mathcal L}(H)}$ is generalized k-quasihyponormal if there exist a constant M>0 such that $T^{\ast k}[M^2(T-z)^{\ast}(T-z)-(T-z)(T-z)^{\ast}]T^k{\geq}0$ for some integer $k{\geq}0$ and all $Z{\in} {\mathbf C}$. In this paper, we show that it T is a generalized k-quasihyponormal operator with the property $0{\not\in}{\sigma}(T)$, then T is subscalar of order 2. As a corollary, we get that such a T has a nontrivial invariant subspace if its spectrum has interior in C.

• Journal of the Korean Mathematical Society
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• v.53 no.1
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• pp.233-246
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• 2016

In this paper, we introduce the class of analytic extensions of M-hyponormal operators and we study various properties of this class. We also use a special Sobolev space to show that every analytic extension of an M-hyponormal operator T is subscalar of order 2k + 2. Finally we obtain that an analytic extension of an M-hyponormal operator satisfies Weyl's theorem.

• Honam Mathematical Journal
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• v.31 no.2
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• pp.137-145
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• 2009

Let T ${\in}$ $\mathcal{L}$(X), S ${\in}$ $\mathcal{L}$(Y ), A ${\in}$ $\mathcal{L}$(X, Y ) and B ${\in}$ $\mathcal{L}$(Y,X) such that SA = AT, TB = BS, AB = S and BA = T. Then S and T shares that same local spectral properties SVEP, property (${\beta}$), property $({\beta})_{\epsilon}$, property (${\delta}$) and decomposability. From these common local spectral properties, we give some results related with Aluthge transforms and subscalar operators.

• Journal of the Chungcheong Mathematical Society
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• v.22 no.2
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• pp.271-279
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• 2009

In this paper, we prove that asymptotically quasisimilar sub-decomposable operators have equal spectra and quasisimilar decomposable operators have equal spectra. Moreover, every subscalar operator is admissible.

• Bulletin of the Korean Mathematical Society
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• v.42 no.3
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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.

• Bulletin of the Korean Mathematical Society
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• v.34 no.1
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• pp.19-28
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• 1997

Let $H$ and $K$ be separable, complex Hilbert spaces and $L(H, K)$ denote the space of all linear, bounded operators from $H$ to $K$. If $H = K$, we write $L(H)$ in place of $L(H, K)$. An operator $T$ in $L(H)$ is called hyponormal if $TT^* \leq T^*T$, or equivalently, if $\left\$\mid$ T^*h \right\$\mid$ \leq \left\$\mid$ Th \right\$\mid$$ for each h in $H$. In [Pu], M. Putinar constructed a universal functional model for hyponormal operators.

• Journal of applied mathematics & informatics
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• v.29 no.1_2
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• pp.459-468
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• 2011

Let T ${\in}$ L(X), S ${\in}$ L(Y), A ${\in}$ L(X, Y) and B ${\in}$ L(Y, X) such that SA = AT, TB = BS, AB = S and BA = T. Then S and T shares the same local spectral properties SVEP, Bishop's property (${\beta}$), property $({\beta})_{\epsilon}$, property (${\delta}$) and and subscalarity. Moreover, the operators ${\lambda}I$ - T and ${\lambda}I$ - S have many basic operator properties in common.

• 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.