• 대한수학회지
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• 제56권3호
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• pp.703-721
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• 2019

The notion of slant H-Toeplitz operator $V_{\phi}$ on the Hardy space $H^2$ is introduced and its characterizations are obtained. It has been shown that an operator on the space $H^2$ is a slant H-Toeplitz if and only if its matrix is a slant H-Toeplitz matrix. In addition, the conditions under which slant Toeplitz and slant Hankel operators become slant H-Toeplitz operators are also obtained.

• 대한수학회보
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• 제58권2호
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• pp.327-347
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• 2021

As an extension to the study of Toeplitz operators on the Bergman space, the notion of H-Toeplitz operators B�� is introduced and studied. Necessary and sufficient conditions under which H-Toeplitz operators become co-isometry and partial isometry are obtained. Some of the invariant subspaces and kernels of H-Toeplitz operators are studied. We have obtained the conditions for the compactness and Fredholmness for H-Toeplitz operators. In particular, it has been shown that a non-zero H-Toeplitz operator can not be a Fredholm operator on the Bergman space. Moreover, we have also discussed the necessary and sufficient conditions for commutativity of H-Toeplitz operators.

• 대한수학회논문집
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• 제35권3호
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• pp.855-866
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• 2020

For two essentially bounded Lebesgue measurable functions 𝜙 and ξ on unit circle 𝕋, we attempt to study properties of operators $S^k_{\mathcal{M}({\phi},{\xi})=S^k_{T_{\phi}}+S^k_{H_{\xi}}$ on H²(𝕋) (k ≥ 2), where $S^k_{T_{\phi}}$ is a k^th-order slant Toeplitz operator with symbol 𝜙 and $S^k_{H_{\xi}}$ is a k^th-order slant Hankel operator with symbol ξ. The spectral properties of operators S^k_{𝓜(𝜙,𝜙)} (or simply S^k_𝓜(𝜙)) are investigated on H²(𝕋). More precisely, it is proved that for k = 2, the Coburn's type theorem holds for S^k_𝓜(𝜙). The conditions under which operators S^k_𝓜(𝜙) commute are also explored.

• 호남수학학술지
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• 제31권4호
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• pp.633-637
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• 2009

We prove using the Szeg$\H{o}$ kernel and the Garabedian kernel that a Toeplitz operator on the boundary of $C^{\infty}$ smoothly bounded domain associated to a smooth symbol vanishes only when the symbol vanishes identically. This gives a generalization of previous results on the unit disk to more general domains in the plane.

• 대한수학회보
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• 제53권6호
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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.

• 대한수학회논문집
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• 제34권4호
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• pp.1229-1243
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• 2019

The notion of $k^{th}$-order essentially slant weighted Toeplitz operator on the weighted Lebesgue space $L^2({\beta})$ is introduced and its algebraic properties are investigated. In addition, the compression of $k^{th}$-order essentially slant weighted Toeplitz operators on the weighted Hardy space $H^2({\beta})$ is also studied.

• 대한수학회논문집
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• 제31권3호
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• pp.585-590
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• 2016

Suppose $T_{\varphi}$ is a 2-hyponormal Toeplitz operator whose self-commutator has rank $n{\geq}1$. If $H_{\bar{\varphi}}(ker[T^*_{\varphi},T_{\varphi}])$ contains a vector $e_n$ in a canonical orthonormal basis $\{e_k\}_{k{\in}Z_+}$ of $H^2({\mathbb{T}})$, then ${\varphi}$ should be an analytic function of the form ${\varphi}=qh$, where q is a finite Blaschke product of degree at most n and h is an outer function.

• 대한수학회논문집
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• 제33권2호
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• pp.507-513
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• 2018

We will consider the asymptotic toeplitzness associated with weighted composition operators on the Hardy-Hilbert space $H^2$.

• 대한수학회논문집
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• 제33권3호
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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.

• 대한수학회보
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• 제51권1호
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• pp.237-252
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• 2014

In this paper, we give a necessary and sufficient condition for the hyponormality of the block Toeplitz operators $T_{\Phi}$, where ${\Phi}$ = $F+G^*$, F(z), G(z) are some matrix valued polynomials on the vector valued Bergman space $L^2_a(\mathbb{D},\mathbb{C}^n)$. We also show some necessary conditions for the hyponormality of $T_{F+G^*}$ with $F+G^*{\in}h^{\infty}{\otimes}M_{n{\times}n}$ on $L^2_a(\mathbb{D},\mathbb{C}^n)$.