• Journal of the Korean Mathematical Society
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• v.54 no.2
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• pp.599-612
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• 2017

If ${\psi}$ is analytic on the open unit disk $\mathbb{D}$ and ${\varphi}$ is an analytic self-map of $\mathbb{D}$, the weighted composition operator $C_{{\psi},{\varphi}}$ is defined by $C_{{\psi},{\varphi}}f(z)={\psi}(z)f({\varphi}(z))$, when f is analytic on $\mathbb{D}$. In this paper, we study normal, cohyponormal, hyponormal and normaloid weighted composition operators on the Hardy and weighted Bergman spaces. First, for some weighted Hardy spaces $H^2({\beta})$, we prove that if $C_{{\psi},{\varphi}}$ is cohyponormal on $H^2({\beta})$, then ${\psi}$ never vanishes on $\mathbb{D}$ and ${\varphi}$ is univalent, when ${\psi}{\not\equiv}0$ and ${\varphi}$ is not a constant function. Moreover, for ${\psi}=K_a$, where |a| < 1, we investigate normal, cohyponormal and hyponormal weighted composition operators $C_{{\psi},{\varphi}}$. After that, for ${\varphi}$ which is a hyperbolic or parabolic automorphism, we characterize all normal weighted composition operators $C_{{\psi},{\varphi}}$, when ${\psi}{\not\equiv}0$ and ${\psi}$ is analytic on $\bar{\mathbb{D}}$. Finally, we find all normal weighted composition operators which are bounded below.

• East Asian mathematical journal
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• v.2
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• pp.87-95
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• 1986

• Honam Mathematical Journal
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• v.32 no.4
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• pp.763-767
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• 2010

In this note, we give a solution of a $H^{\infty}$-optimization problem.

• 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.

• Journal of the Korean Mathematical Society
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• v.45 no.4
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• pp.1027-1041
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• 2008

In this paper we consider the hyponormality of Toeplitz operators $T_{\varphi}$ on the Bergman space $L_a^2{(\mathbb{D})$ in the cases, where ${\varphi}\;:=f+\bar{g}$ (f and g are polynomials). We present some necessary or sufficient conditions for the hyponormality of $T_{\varphi}$ under certain assumptions about the coefficients of ${\varphi}$.

• Korean Journal of Mathematics
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• v.15 no.2
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• pp.185-193
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• 2007

In this paper we consider the hyponormality of Toeplitz operators $T_{\varphi}$ on the Bergman space $L^2_a({\mathbb{D})$ with symbol in the case of function $f+{\overline{g}}$ with polynomials $f$ and $g$. We present some necessary conditions for the hyponormality of $T_{\varphi}$ under certain assumptions about the coefficients of ${\varphi}$.

• Communications of the Korean Mathematical Society
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• v.26 no.1
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• pp.79-88
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• 2011

In this paper we study some properties of rank one perturbations of the unilateral shift operators $T=S+u{\otimes}{\upsilon}$. In particular, we give some criteria for eigenvalues of T. Also we characterize some conditions for T to be hyponormal.

• Bulletin of the Korean Mathematical Society
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• v.54 no.2
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• pp.633-646
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• 2017

Recently, Curto, Lee and Yoon considered the properties (such as, hyponormality, subnormality, and flatness, etc.) for 2-variable weighted shifts and constructed several families of commuting pairs of subnormal operators such that each family can be used to answer a conjecture of Curto, Muhly and Xia negatively. In this paper, we consider the weak and quadratic hyponormality of 2-variable weighted shifts ($W_1,W_2$). In addition, we detect the weak and quadratic hyponormality with some interesting 2-variable weighted shifts.

• Communications of the Korean Mathematical Society
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• v.33 no.3
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• pp.787-798
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• 2018

For $1{\leq}p{\leq}{\infty}$, let $H^p$ be the usual Hardy space on the unit circle. When ${\alpha}$ and ${\beta}$ are bounded functions, a singular integral operator $S_{{\alpha},{\beta}}$ is defined as the following: $S_{{\alpha},{\beta}}(f+{\bar{g}})={\alpha}f+{\beta}{\bar{g}}(f{\in}H^p,\;g{\in}zH^p)$. When p = 2, we study the hyponormality of $S_{{\alpha},{\beta}}$ when ${\alpha}$ and ${\beta}$ are some special functions.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.12 no.1
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• pp.1-6
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• 2008

Duggal-Jeon-Kubrusly([2]) introduced Hilbert space operator T satisfying property ${\mid}T{\mid}^2{\leq}{\mid}T^2{\mid}$, where ${\mid}T{\mid}=(T^*T)^{1/2}$. In this paper we extend this property to general version, namely property B(n). In addition, we construct examples which distinguish the classes of operators with property B(n) for each $n{\in}\mathbb{N}$.