• Journal of applied mathematics & informatics
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• v.36 no.3_4
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• pp.237-244
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• 2018

Let ${\mathcal{H}}$ be an infinite dimensional separable Hilbert space with a fixed orthonormal base $\{e_1,e_2,{\cdots}\}$. Let L be the subspace lattice generated by the subspaces $\{[e_1],[e_1,e_2],[e_1,e_2,e_3],{\cdots}\}$ and let $Alg{\mathcal{L}}$ be the algebra of bounded operators which leave invariant all projections in ${\mathcal{L}}$. Let p and q be natural numbers (p < q). Let ${\mathcal{A}}$ be a linear manifold in $Alg{\mathcal{L}}$ such that $T_{(p,q)}=0$ for all T in ${\mathcal{A}}$. If ${\mathcal{A}}$ is a Lie ideal, then $T_{(p,p)}=T_{(p+1,p+1)}={\cdots}=T_{(q,q)}$ and $T_{(i,j)}=0$, $p{\eqslantless}i{\eqslantless}q$ and i < $j{\eqslantless}q$ for all T in ${\mathcal{A}}$.

• Honam Mathematical Journal
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• v.31 no.2
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• pp.259-265
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• 2009

Let $\mathcal{L}$ be a subspace lattice on a Hilbert space $\mathcal{H}$. Let x and y be vectors in $\mathcal{H}$ and let $P_x$ be the projection onto sp(x). If $P_xE$ = $EP_x$ for each E ${\in}\;\mathcal{L}$, then the following are equivalent. (1) There exists an operator A in Alg$\mathcal{L}$ such that Ax = y, Af = 0 for all f in $sp(x)^{\perp}$ and A ${\geq}$ 0. (2) sup ${\frac{{\parallel}E^{\perp}y{\parallel}}{{\parallel}E^{\perp}x{\parallel}}:E{\in}\mathcal{L}}$ < ${\infty}$ < x, y > ${\geq}$ 0. Let X and Y be operators in $\mathcal{B}(\mathcal{H})$. Let P be the projection onto $\overline{rangeX}$. If PE = EP for each E ${\in}\;\mathcal{L}$, then the following are equivalent: (1) sup ${\frac{{\parallel}E^{\perp}Yf{\parallel}}{{\parallel}E^{\perp}Xf{\parallel}}:f{\in}\mathcal{H},E{\in}\mathcal{L}}$ < ${\infty}$ and < Xf, Yf > ${\geq}$ 0 for all f in H. (2) There exists a positive operator A in Alg$\mathcal{L}$ such that AX = Y.

• Honam Mathematical Journal
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• v.27 no.2
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• pp.243-250
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• 2005

Given operators X and Y acting on a separable Hilbert space ${\mathcal{H}}$, an interpolating operator is a bounded operator A such that AX = Y. We show the following: Let ${\mathcal{L}}$ be a subspace lattice acting on a separable complex Hilbert space ${\mathcal{H}}$. and let $X=(x_{ij})$ and $Y=(y_{ij})$ be operators acting on ${\mathcal{H}}$. Then the following are equivalent: (1) There exists an invertible operator $A=(a_{ij})$ in $Alg{\mathcal{L}}$ such that AX = Y. (2) There exist bounded sequences {${\alpha}_n$} and {${\beta}_n$} in ${\mathbb{C}}$ such that $${\alpha}_{2k-1}{\neq}0,\;{\beta}_{2k-1}=\frac{1}{{\alpha}_{2k-1}},\;{\beta}_{2k}=-\frac{{\alpha}_{2k}}{{\alpha}_{2k-1}{\alpha}_{2k+1}}$$ and $$y_{i1}={\alpha}_1x_{i1}+{\alpha}_2x_{i2}$$ $$y_{i\;2k}={\alpha}_{4k-1}x_{i\;2k}$$ $$y_{i\;2k+1}={\alpha}_{4k}x_{i\;2k}+{\alpha}_{4k+1}x_{i\;2k+1}+{\alpha}_{4k+2}x_{i\;2k+2}$$ for $$k{\in}N$$.

• Honam Mathematical Journal
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• v.39 no.4
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• pp.503-514
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• 2017

For real parameters ${\alpha}$ and ${\beta}$ such that ${\alpha}$ < 1 < ${\beta}$, we denote by $\mathcal{P}({\alpha},{\beta})$ the class of analytic functions p, which satisfy p(0) = 1 and ${\alpha}$ < ${\Re}\{p(z)\}$ < ${\beta}$ in ${\mathbb{D}}$, where ${\mathbb{D}}$ denotes the open unit disk. Let ${\mathcal{A}}$ be the class of analytic functions in ${\mathbb{D}}$ such that f(0) = 0 = f'(0) - 1. For $f{\in}{\mathcal{A}}$, ${\mu}{\in}{\mathbb{C}}{\backslash}\{0\}$ and ${\nu}{\in}{\mathbb{C}}$, let $I_{{\mu},{\nu}:{\mathcal{A}}{\rightarrow}{\mathcal{A}}$ be an integral operator defined by $$I_{{\mu},{\nu}[f](z)}=\({\frac{{\mu}+{\nu}}{z^{\nu}}}{\int}^z_0f^{\mu}(t)t^{{\nu}-1}dt\)^{1/{\mu}}$$. In this paper, we find some sufficient conditions on functions to be in the class $\mathcal{P}({\alpha},{\beta})$. One of these results is applied to the integral operator $I_{{\mu},{\nu}}$ of two classes of starlike functions which are related to the class $\mathcal{P}({\alpha},{\beta})$.

• Honam Mathematical Journal
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• v.37 no.3
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• pp.281-285
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• 2015

An A-m-isometric operator is a bounded linear operator T on a Hilbert space $\mathcal{H}$ satisfying $\sum\limits_{k=0}^{m}(-1)^{m-k}T^{*^k}AT^k=0$, where A is a positive operator. We give sufficient conditions under which A-m-isometries are not N-supercyclic, for every $N{\geq}1$; that is, there is not a subspace E of dimension N such that its orbit under T is dense in $\mathcal{H}$.

• Korean Journal of Mathematics
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• v.19 no.2
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• pp.205-209
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• 2011

Let $\mathcal{QA}$ denote the class of bounded linear Hilbert space operators T which satisfy the operator inequality $T^*|T^2|T{\geq}T^*|T|^2T$. Let $f$ be an analytic function defined on an open neighbourhood $\mathcal{U}$ of ${\sigma}(T)$ such that $f$ is non-constant on the connected components of $\mathcal{U}$. We generalize a theorem of Sheth [10] to $f(T){\in}\mathcal{QA}$.

• Bulletin of the Korean Mathematical Society
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• v.52 no.4
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• pp.1269-1283
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• 2015

An operator $T{{\in}}{\mathcal{L}}({\mathcal{H}})$ is said to be skew complex symmetric if there exists a conjugation C on ${\mathcal{H}}$ such that $T=-CT^*C$. In this paper, we study properties of skew complex symmetric operators including spectral connections, Fredholmness, and subspace-hypercyclicity between skew complex symmetric operators and their adjoints. Moreover, we consider Weyl type theorems and Browder type theorems for skew complex symmetric operators.

• Honam Mathematical Journal
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• v.32 no.2
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• pp.255-260
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• 2010

Given vectors x and y in a separable complex Hilbert space $\mathcal{H}$, an interpolating operator is a bounded operator A such that Ax = y. In this article, we investigate compact interpolation problems for vectors in a tridiagonal algebra. We show the following : Let Alg$\mathcal{L}$ be a tridiagonal algebra on a separable complex Hilbert space $\mathcal{H}$ and let x = $(x_i)$ and y = $(y_i)$ be vectors in H. Then the following are equivalent: (1) There exists a compact operator A = $(a_{ij})$ in Alg$\mathcal{L}$ such that Ax = y. (2) There is a sequence ${{\alpha}_n}$ in $\mathbb{C}$ such that ${{\alpha}_n}$ converges to zero and for all k ${\in}$ $\mathbb{N}$, $y_1 = {\alpha}_1x_1 + {\alpha}_2x_2$ $y_{2k} = {\alpha}_{4k-1}x_{2k}$ $y_{2k+1}={\alpha}_{4k}x_{2k}+{\alpha}_{4k+1}x_{2k+1}+{\alpha}_{4k+2}+x_{2k+2}$.

• Journal of Integrative Natural Science
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• v.10 no.2
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• pp.115-118
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• 2017

We study a notion of $\mathcal{A}$-frequent hypercyclicity of linear maps between the Banach algebras consisting of operators on a separable infinite dimensional Banach space. We prove a sufficient condition for a linear map to satisfy the $\mathcal{A}$-frequent hypercyclicity in the strong operator topology.

• Bulletin of the Korean Mathematical Society
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• v.52 no.4
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• pp.1225-1240
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• 2015

We provide a complete proof that there are no eigenvalues of the integral operator ${\mathcal{K}}_l$ outside the interval (0, 1/k). ${\mathcal{K}}_l$ arises naturally from the deflection problem of a beam with length 2l resting horizontally on an elastic foundation with spring constant k, while some vertical load is applied to the beam.