• 대한수학회지
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• 제46권6호
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• pp.1219-1232
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• 2009

Let H(B) denote the space of all holomorphic functions on the unit ball B of $\mathbb{C}^n$. Let $\varphi$ = (${\varphi}_1,{\ldots}{\varphi}_n$) be a holomorphic self-map of B and $g{\in}2$(B) with g(0) = 0. In this paper we study the boundedness and compactness of the generalized composition operator $C_{\varphi}^gf(z)=\int_{0}^{1}{\mathfrak{R}}f(\varphi(tz))g(tz){\frac{dt}{t}}$ from generalized weighted Bergman spaces into Bloch type spaces.

• 대한수학회보
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• 제54권4호
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• pp.1221-1228
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• 2017

We consider weighted Hardy spaces on bidisk ${\mathbb{D}}^2$ which generalize the weighted Bergman spaces $A^2_{\alpha}({\mathbb{D}}^2)$. Let z, w be coordinate functions and $T_{{\bar{z}}^N}_w$ Toeplitz operator with symbol $_{{\bar{z}}^N}_w$. In this paper, we study the reducing subspaces of $T_{{\bar{z}}^N}_w$ on the weighted Hardy spaces.

• 대한수학회지
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• 제48권5호
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• pp.1043-1052
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• 2011

We introduce a new norm, called the $N^p$-norm (1 $\leq$ p < ${\infty}$ on the space $N^p$(V,W) where V and W are abstract operator spaces. By proving some fundamental properties of the space $N^p$(V,W), we also discover that if W is complete, then the space $N^p$(V,W) is also a Banach space with respect to this norm for 1 $\leq$ p < ${\infty}$.

• 대한수학회지
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• 제54권2호
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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.

• 대한수학회논문집
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• 제28권2호
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• pp.285-295
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• 2013

In this note we present sufficient conditions for the existence of Radon-Nikodym derivatives (RND) of operator valued measures with respect to scalar measures. The RND is characterized by the Bochner integral in the strong operator topology of a strongly measurable operator valued function with respect to a nonnegative finite measure. Using this result we also obtain a characterization of compact sets in the space of operator valued measures. An extension of this result is also given using the theory of Pettis integral. These results have interesting applications in the study of evolution equations on Banach spaces driven by operator valued measures as structural controls.

• Nonlinear Functional Analysis and Applications
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• 제28권1호
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• pp.265-286
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• 2023

In this paper, we introduce and study a generalized Cayley operator associated to H(·, ·)-monotone operator in semi-inner product spaces. Using the notion of graph convergence, we give the equivalence result between graph convergence and convergence of generalized Cayley operator for the H(·, ·)-monotone operator without using the convergence of the associated resolvent operator. To support our claim, we construct a numerical example. As an application, we consider a system of generalized Cayley inclusions involving H(·, ·)-monotone operators and give the existence and uniqueness of the solution for this system. Finally, we propose a perturbed iterative algorithm for finding the approximate solution and discuss the convergence of iterative sequences generated by the perturbed iterative algorithm.

• 대한수학회보
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• 제59권3호
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• pp.659-669
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• 2022

In this paper we introduce the notion of universal free product for operator systems and operator spaces, and prove extension results for the operator system lifting property (OSLP) and operator system local lifting property (OSLLP) to the universal free product.

• 대한수학회보
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• 제52권4호
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• pp.1383-1399
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• 2015

Let ${\mathbb{D}}=\{z{\in}{\mathbb{C}}:{\mid}z{\mid}<1\}$ be the open unit disk in the complex plane $\mathbb{C}$, ${\varphi}$ an analytic self-map of $\mathbb{D}$ and ${\psi}$ an analytic function in $\mathbb{D}$. Let D be the differentiation operator and $W_{{\varphi},{\psi}}$ the weighted composition operator. The boundedness and compactness of the product-type operator $W_{{\varphi},{\psi}}D$ from the weighted Bergman-Orlicz space to the weighted Zygmund space on $\mathbb{D}$ are characterized.

• 대한수학회지
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• 제41권1호
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• pp.95-105
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• 2004

In this paper we will consider the weighted composition operators W = $uC_{\tau}$ between $L^{p}$$(X,\sum,\mu$) spaces and Orlicz spaces $L^{\phi}$$(X,\sum,\mu$) generated by measurable and non-singular transformations $\tau$ from X into itself and measurable functions u on X. We characterize the functions u and transformations $\tau$ that induce weighted composition operators between $L^{p}$ -spaces by using some properties of conditional expectation operator, pair (u,${\gamma}$) and the measure space $(X,\sum,\mu$). Also, some other properties of these types of operators will be investigated.

• 대한수학회지
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• 제45권1호
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• pp.229-248
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• 2008

The boundedness and compactness of the Volterra composition operators between weighted Bergman spaces and Bloch type spaces are discussed in this paper.