• 대한수학회지
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• 제52권2호
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• pp.403-416
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• 2015

An operator T on a complex Hilbert space $\mathcal{H}$ is called skew symmetric if T can be represented as a skew symmetric matrix relative to some orthonormal basis for $\mathcal{H}$. In this note, we explore the structure of skew symmetric operators with disconnected spectra. Using the classical Riesz decomposition theorem, we give a decomposition of certain skew symmetric operators with disconnected spectra. Several corollaries and illustrating examples are provided.

• East Asian mathematical journal
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• 제31권5호
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• pp.695-706
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• 2015

Let ${\mathcal{H}}$ be an infinite dimensional complex Hilbert space, and let $B({\mathcal{H}})$ be the algebra of all bounded linear operators on ${\mathcal{H}}$. Recall that an operator $T{\in}B({\mathcal{H})$ has property B(n) if ${\mid}T^n{\mid}{\geq}{\mid}T{\mid}^n$, $n{\geq}2$, which generalizes the class A-operator. We characterize the property B(n) of weighted shifts $S_{\lambda}$ over (${\eta},\;{\kappa}$)-type directed trees which appeared in the study of subnormality of weighted shifts over directed trees recently. In addition, we discuss the property B(n) of weighted shifts $S_{\lambda}$ over (2, 1)-type directed trees with nonzero weights are being distinct with respect to $n{\geq}2$. And we give some properties of weighted shifts $S_{\lambda}$ over (2, 1)-type directed trees with property B(2).

• 대한수학회보
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• 제49권3호
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• pp.481-493
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• 2012

Let $S$ be a upper triangular operator such that $M^L_S:\mathcal{U}_2{\rightarrow}\mathcal{U}_2$ defined by $M^L_S(F)=SF$ is a contraction. Then there exists an unitary linear system whose state space is the extension space $\tilde{D}_L$(S) associated with $\mathcal{H}_L$(S).

• 대한수학회보
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• 제49권5호
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• pp.899-910
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• 2012

Let $\mathcal{H}$ be a complex separable infinite dimensional Hilbert space. In this paper, a necessary and sufficient condition is given for an operator T on $\mathcal{H}$ to satisfy that $f(T)$ obeys generalized Weyl's theorem for each function $f$ analytic on some neighborhood of ${\sigma}(T)$. Also we investigate the stability of generalized Weyl's theorem under (small) compact perturbations.

• Korean Journal of Mathematics
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• 제16권3호
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• pp.349-353
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• 2008

Let the set of all quasi-class A operators for which $ker(A){\subseteq}ker(A^*)$ be denoted by $A{\in}{\mathcal{Q}}A^*$. In this paper it is proved that an operator $T{\in}{\mathcal{Q}}A^*$ is normal if and only if the Duggal transform of T is normal.

• 대한수학회보
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• 제52권2호
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• pp.581-591
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• 2015

Let $\mathcal{A}$ be a factor von Neumann algebra with dimension greater than 1. We prove that if a linear map ${\delta}:\mathcal{A}{\rightarrow}\mathcal{A}$ satisfies $${\delta}([[a,b],c])=[[{\delta}(a),b],c]+[[a,{\delta}(b),c]+[[a,b],{\delta}(c)]$$ for any $a,b,c{\in}\mathcal{A}$ with ab = 0 (resp. ab = P, where P is a fixed nontrivial projection of $\mathcal{A}$), then there exist an operator $T{\in}\mathcal{A}$ and a linear map $f:\mathcal{A}{\rightarrow}\mathbb{C}I$ vanishing at every second commutator [[a, b], c] with ab = 0 (resp. ab = P) such that ${\delta}(a)=aT-Ta+f(a)$ for any $a{\in}\mathcal{A}$.

• 대한수학회보
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• 제52권6호
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• pp.1963-1971
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• 2015

Let $\mathcal{A}$ and $\mathcal{B}$ be two unital $C^*$-algebras. Denote by W(a) the numerical range of an element $a{\in}\mathcal{A}$. We show that the condition W(ax) = W(bx), ${\forall}x{\in}\mathcal{A}$ implies that a = b. Using this, among other results, it is proved that if ${\phi}$ : $\mathcal{A}{\rightarrow}\mathcal{B}$ is a surjective map such that $W({\phi}(a){\phi}(b){\phi}(c))=W(abc)$ for all a, b and $c{\in}\mathcal{A}$, then ${\phi}(1){\in}Z(B)$ and the map ${\psi}={\phi}(1)^2{\phi}$ is multiplicative.

• 충청수학회지
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• 제25권4호
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• pp.655-668
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• 2012

An operator $T\;{\varepsilon}\;B(\mathcal{H})$ is said to be $k$-quasi-paranormal operator if $||T^{k+1}x||^2\;{\leq}\;||T^{k+2}x||\;||T^kx||$ for every $x\;{\epsilon}\;\mathcal{H}$, $k$ is a natural number. This class of operators contains the class of paranormal operators and the class of quasi - class A operators. In this paper, using the operator matrix representation of $k$-quasi-paranormal operators which is related to the paranormal operators, we show that every algebraically $k$-quasi-paranormal operator has Bishop's property ($\beta$), which is an extension of the result proved for paranormal operators in [32]. Also we prove that (i) generalized Weyl's theorem holds for $f(T)$ for every $f\;{\epsilon}\;H({\sigma}(T))$; (ii) generalized a - Browder's theorem holds for $f(S)$ for every $S\;{\prec}\;T$ and $f\;{\epsilon}\;H({\sigma}(S))$; (iii) the spectral mapping theorem holds for the B - Weyl spectrum of T.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제11권1호
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• pp.29-36
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• 2004

Given vectors x and ｙ in a Hilbert space, an interpolating operator is a bounded operator T such that Tx=y. In this paper the following is proved: Let $\cal{L}$ be a subspace lattice on a Hilbert space $\cal{H}$. Let x and ｙ be vectors in $\cal{H}$ and let $P_x$, be the projection onto sp(x). If $P_xE=EP_x$ for each $ E \in \cal{L}$ then the following are equivalent. (1) There exists an operator A in Alg(equation omitted) such that Ax=y, Af = 0 for all f in ($sp(x)^\perp$) and $A=-A^\ast$. (2) (equation omitted)

• East Asian mathematical journal
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• 제16권2호
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• pp.169-174
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• 2000

We give an easy proof that ${\phi}(z)=\frac{z^2_1}{1-z^2_2}$ induces the bounded composition operator $C_{\phi}:\mathcal{B}{\rightarrow}{\bigcap}H^p(B_2)$ defined by $C_{\phi}f=f\;o\;{\phi}$.