• 한국전산응용수학회:학술대회논문집
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• 한국전산응용수학회 2003년도 KSCAM 학술발표회 프로그램 및 초록집
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• pp.4.1-4
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• 2003

Given operators X and Y acting on a Hilbert space H, an interpolating operator is a bounded operator A such that AX= Y. An interpolating operator for n-operators satisfies the equation AXi= Yi, for i = 1,2,...,n, In this article, we showed the following : Let H be a Hilbert space and let L be a subspace lattice on H. Let X and Y be operators acting on H. Assume that rangeX is dense in H. Then the following statements are equivalent : (1) There exists an operator A in AlgL such that AX = Y, A$\^$*/=A and every E in L reduces A. (2) sup｛(equation omitted) : n $\in$ N f$\sub$I/ $\in$ H and E$\sub$I/ $\in$ L｝<$\infty$ and = for all E in L and all f, g in H.

• 대한수학회지
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• 제54권2호
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• pp.563-574
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• 2017

Let $\mathcal{A}$ be a unital real standard Jordan operator algebra acting on a Hilbert space H of dimension at least 2. We show that every bijection ${\phi}$ on $\mathcal{A}$ satisfying ${\phi}(A^2{\circ}B)={\phi}(A)^2{\circ}{\phi}(B)$ is of the form ${\phi}={\varepsilon}{\psi}$ where ${\psi}$ is an automorphism on $\mathcal{A}$ and ${\varepsilon}{\in}\{-1,1\}$. As a consequence if $\mathcal{A}$ is the real algebra of all self-adjoint operators on a Hilbert space H, then there exists a unitary or conjugate unitary operator U on H such that ${\phi}(A)={\varepsilon}UAU^*$ for all $A{\in}\mathcal{A}$.

• Kyungpook Mathematical Journal
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• 제55권4호
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• pp.885-892
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• 2015

In this paper, we mainly obtain the following assertions: (1) If T is a quasi-*-n-paranormal operator, then T is finite and simply polaroid. (2) If T or $T^*$ is a quasi-*-n-paranormal operator, then Weyl's theorem holds for f(T), where f is an analytic function on ${\sigma}(T)$ and is not constant on each connected component of the open set U containing ${\sigma}(T)$. (3) If E is the Riesz idempotent for a nonzero isolated point ${\lambda}$ of the spectrum of a quasi-*-n-paranormal operator, then E is self-adjoint and $EH=N(T-{\lambda})=N(T-{\lambda})^*$.

• 대한수학회보
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• 제46권1호
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• pp.117-123
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• 2009

This paper is a study of some properties of a collection of bounded linear operators resulting from terraced matrices M acting through multiplication on ${\ell}^2$; the term terraced matrix refers to a lower triangular infinite matrix with constant row segments. Sufficient conditions are found for M to be posinormal, meaning that $MM^*=M^*PM$ for some positive operator P on ${\ell}^2$; these conditions lead to new sufficient conditions for the hyponormality of M. Sufficient conditions are also found for the adjoint $M^*$ to be posinormal, and it is observed that, unless M is essentially trivial, $M^*$ cannot be hyponormal. A few examples are considered that exhibit special behavior.

• 대한수학회지
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• 제48권5호
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• pp.969-984
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• 2011

This paper discusses the hypercyclicity of weighted composition operators acting on the space of holomorphic functions on the open unit ball $B_N$ of $\mathbb{C}^N$. Several analytic properties of linear fractional self-maps of $B_N$ are given. According to these properties, a few necessary conditions for a weighted composition operator to be hypercyclic in the space of holomorphic functions are proved. Besides, the hypercyclicity of adjoint of weighted composition operators are studied in this paper.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제18권1호
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• pp.31-39
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• 2011

Stone's theorem states that "A bounded linear operator A is infinitesimal generator of a $C_0$-group of unitary operators on a Hilbert space H if and only if iA is self adjoint". In this paper we establish a generalization of Stone's theorem in the framework of Hilbert $C^*$-modules.

• 대한수학회보
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• 제58권1호
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• pp.235-251
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• 2021

We consider the Schrödinger type operator ��k = (-∆)k+Vk on ℝn(n ≥ 2k + 1), where k = 1, 2 and the nonnegative potential V belongs to the reverse Hölder class RHs with n/2 < s < n. In this paper, we establish the (Lp, Lq)-boundedness of the higher order Riesz transform T��,�� = V2��∇2��-��2 (0 ≤ �� ≤ 1/2 < �� ≤ 1, �� - �� ≥ 1/2) and its adjoint operator T∗��,�� respectively. We show that T��,�� is bounded from Hardy type space $H^1_{\mathcal{L}_2}({\mathbb{R}}_n)$ into Lp2 (ℝn) and T∗��,�� is bounded from ��p1 (ℝn) into BMO type space $BMO_{\mathcal{L}_1}$ (ℝn) when �� - �� > 1/2, where $p_1={\frac{n}{4({\beta}-{\alpha})-2}}$, $p_2={\frac{n}{n-4({\beta}-{\alpha})+2}}$. Moreover, we prove that T��,�� is bounded from $BMO_{\mathcal{L}_1}({\mathbb{R}}_n)$ to itself when �� - �� = 1/2.

• 대한수학회논문집
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• 제38권4호
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• pp.1127-1139
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• 2023

We investigate an extension of Schauder's theorem by studying the relationship between various s-numbers of an operator T and its adjoint T^*. We have three main results. First, we present a new proof that the approximation number of T and T^* are equal for compact operators. Second, for non-compact, bounded linear operators from X to Y, we obtain a relationship between certain s-numbers of T and T^* under natural conditions on X and Y . Lastly, for non-compact operators that are compact with respect to certain approximation schemes, we prove results for comparing the degree of compactness of T with that of its adjoint T^*.

• Korean Journal of Mathematics
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• 제24권2호
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• pp.273-296
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• 2016

Some new trace inequalities for convex functions of self-adjoint operators in Hilbert spaces are provided. The superadditivity and monotonicity of some associated functionals are investigated. Some trace inequalities for matrices are also derived. Examples for the operator power and logarithm are presented as well.

• Korean Journal of Mathematics
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• 제31권3호
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• pp.313-321
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• 2023

In this paper, we explore the additivity of the map Ω : 𝒜 → 𝒜 that satisfies Ω([ℒ, ℳ]_*)=[Ω (ℳ), ℒ]_* + [ℳ, Ω(ℒ)]_*, where [ℒ, ℳ]_*= ℒℳ^* - ℳ ℒ^*, for all ℒ, ℳ ∈ 𝒜, a prime *-algebra with unit ℐ. Additionally we show that if Ω (αℐ) is self-adjoint operator for α ∈ {1, i} then Ω = 0.