• Bulletin of the Korean Mathematical Society
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• v.53 no.6
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• pp.1823-1830
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• 2016

Let H and K be two Hilbert spaces, and let A and B be two bounded linear operators from H to K. We are interested in $RangeB^*{\supseteq}RangeA^*$. It is well known that this is equivalent to the inequality $A^*A{\geq}{\varepsilon}B^*B$ for a positive constant ${\varepsilon}$. We study conditions in terms of symbols when A and B are singular integral operators, Hankel operators or Toeplitz operators, etc.

• Journal of the Korean Mathematical Society
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• v.45 no.5
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• pp.1417-1425
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• 2008

In this paper we consider the solvability and describe the set of the solutions of the operator equations $AX+X^{*}C=B$ and $AXB+B^{*}X^{*}A^{*}=C$. This generalizes the results of D. S. Djordjevic [Explicit solution of the operator equation $A^{*}X+X^{*}$A=B, J. Comput. Appl. Math. 200(2007), 701-704].

• Bulletin of the Korean Mathematical Society
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• v.35 no.2
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• pp.319-324
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• 1998

The equation AX = BX implies $A^*X\;=\;B^X$ when A and B are normal (Fuglede-Putnam theorem). In this paper, the hypotheses on A and B can be relaxed by usin a Hilbert-Schmidt operator X: Let A be p-quasihyponormal and let $B^*$ be invertible p-quasihyponormal such that AX = XB for a Hilbert-Schmidt operator X and $|||A^*|^{1-p}||{\cdot}|||B^{-1}|^{1-p}||\;{\leq}\;1$.Then $A^*X\;=\;XB^*$.

• Bulletin of the Korean Mathematical Society
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• v.43 no.4
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• pp.747-753
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• 2006

We say operators A, B on Hilbert space satisfy Fuglede-Putnam theorem if AX = X B for some X implies $A^*X=XB^*$. We show that if either (1) A is p-hyponormal and $B^*$ is a class y operator or (2) A is a class y operator and $B^*$ is p-hyponormal, then A, B satisfy Fuglede-Putnam theorem.

• Honam Mathematical Journal
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• v.23 no.1
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• pp.51-57
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• 2001

Let $\mathcal{B}(H)$ and $\mathcal{B}(K)$ denote the algebras of all bounded linear operators on Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$, respectively. We show that if ${\phi}:\mathcal{B}(H){\rightarrow}\mathcal{B}(K)$ is an additive mapping satisfying ${\phi}({\mid}A{\mid}^2)={\mid}{\phi}(A){\mid}^2$ for every $A{\in}\mathcal{B}(H)$, then there exists a mapping ${\psi}$ defined by ${\psi}(A)={\phi}(I){\phi}(A)$, ${\forall}A{\in}\mathcal{B}(H)$ such that ${\psi}$ is the sum of $two^*$-homomorphisms one of which C-linear and the othere C-antilinear. We will also study some conditions implying the injective and rank-preserving of ${\psi}$.

• The Pure and Applied Mathematics
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• v.10 no.4
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• pp.273-277
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• 2003

The Fuglede-Putnam Theorem is that if A and B are normal operators and X is an operator such that AX = XB, then $A^{\ast}= X. In this paper, we show that if A is $\omega$-hyponormal and $B^{\ast}$ is invertible $\omega$-hyponormal such that AX = XB for a Hilbert-Schmidt operator X, then $A^{\ast}X = XB^{\ast}$.

• Communications of the Korean Mathematical Society
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• v.12 no.4
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• pp.895-904
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• 1997

Let B(z) be a power series with operator coefficients where multiplication by B(z), T, is a contractive and everywhere defined transforamtion in the square summable power series. Then there is a Julia operator U for T such that $$ U = (T D)(\tilde{D}^* L) \in B(H \oplus D, K \oplus \tilde{D}), $$ where D is the state space of a conjugate canonical linear system with transfer function B(z).

• Bulletin of the Korean Mathematical Society
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• v.61 no.4
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• pp.1053-1066
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• 2024

In this paper, we study the boundedness of the commutator H^b_Ω formed by the rough Hardy operator H_Ω and a locally integrable function b from homogeneous Herz spaces to homogeneous weak Herz spaces. In addition, the weak boundedness of H^b_Ω on central Morrey spaces is also established.

• Bulletin of the Korean Mathematical Society
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• v.59 no.3
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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.

• Communications of the Korean Mathematical Society
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• v.21 no.1
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• pp.65-74
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• 2006

In this paper, we will define the Bergman projection type operator Pr and find conditions on which the operator Pr is bound-ed on $L^p$(B, dv). By using the properties of the Bergman projection type operator Pr, we will show that if $f{\in}L_a^p$(B, dv), then $(1-{\parallel}{\omega}{\parallel}^2){\nabla}f(\omega){\cdot}z{\in}L^p(B,dv)$. We will also show that if $(1-{\parallel}{\omega}{\parallel}^2)\; \frac{{\nabla}f(\omega){\cdot}z}{},\;{\in}L^p{B,\;dv),\;then\;f{\in}L_a^p(B,\;dv)$.