• Communications of the Korean Mathematical Society
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• v.38 no.3
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• pp.821-835
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• 2023

Hankel operators and their variants have abundant applications in numerous fields. For a non-zero complex number r, the r-Hankel operators on a Hilbert space 𝓗 define a class of one such variant. This article introduces and explores some properties of two other variants of Hankel operators namely k^th-order (C, r)-Hankel operators and k^th-order (R, r)-Hankel operators (k ≥ 2) which are closely related to r-Hankel operators in such a way that a k^th-order (C, r)-Hankel matrix is formed from r^k-Hankel matrix on deleting every consecutive (k - 1) columns after the first column and a k^th-order (R, r^k)-Hankel matrix is formed from r-Hankel matrix if after the first column, every consecutive (k - 1) columns are deleted. For |r| ≠ 1, the characterizations for the boundedness of these operators are also completely investigated. Finally, an appropriate approach is also presented to extend these matrices to two-way infinite matrices.

• Bulletin of the Korean Mathematical Society
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• v.56 no.1
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• pp.187-200
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• 2019

Truncated Hankel operators are compressions of classical Hankel operators to model spaces. In this paper we describe matrix representations of truncated Hankel operators on finite-dimensional model spaces. We then show that the obtained descriptions hold also for some infinite-dimensional cases.

• Communications of the Korean Mathematical Society
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• v.37 no.1
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• pp.91-103
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• 2022

In this paper, we introduce Toeplitz operators and Hankel operators with complex Borel measures on the closed unit disk. When a positive measure 𝜇 on (-1, 1) is a Carleson measure, it is known that the corresponding Hankel matrix is bounded and vice versa. We show that for a positive measure 𝜇 on 𝔻, 𝜇 is a Carleson measure if and only if the Toeplitz operator with symbol 𝜇 is a densely defined bounded linear operator. We also study Hankel operators of Hilbert-Schmidt class.

• Honam Mathematical Journal
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• v.30 no.1
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• pp.147-164
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• 2008

In the setting of the half-plane of the complex plane, we show that for r > $-{\frac{1}{2}}$, the dual space of the weighted Bergman spaces $B^{1,r}$ is the Bloch space and we introduce some operators. We also study Toeplitz and Hankel operators and we find some characterization of Hankel operators.

• Honam Mathematical Journal
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• v.43 no.1
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• pp.59-67
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• 2021

We characterize pluriharmonic symbols of the finite rank little Hankel operators on the Dirichlet space of the unit polydisk.

• Journal of Institute of Control, Robotics and Systems
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• v.6 no.1
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• pp.1-7
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• 2000

This paper studies the delay-characteristics using the singular values and vectors of Hankel operators for input delay systems. First, the computational method of Hankel singular values and their corresponding singular vectors are introduced, and then it is analytically provea that all the Hankel singular vlues have a monotone increasing properties as the length of delay time increases. Furthermore, through a simple numerical example, it is shown that the Hankel singular values are dependent only on the ratio of the time constant of a lumped parameter system to the length of delay , and in case that the time constant is relatively larger than the delay time, the lumped parameter characteristic has a great influence on the input delay systems.

• Journal of the Korean Mathematical Society
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• v.56 no.3
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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.

• 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.

• Communications of the Korean Mathematical Society
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• v.35 no.3
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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.

• Communications of the Korean Mathematical Society
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• v.33 no.4
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• pp.1249-1269
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• 2018

This paper provides a method to study the nonnegativity of certain linear operators, from other operators with similar spectral properties. If these new operators are formally self-adjoint and nonnegative, we can study the complex powers using an appropriate locally convex space. In this case, the initial operator also will be nonnegative and we will be able to study its powers. In particular, we have applied this method to Bessel-type operators.