• Title/Summary/Keyword: Hankel operator

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GENERALIZED (C, r)-HANKEL OPERATOR AND (R, r)-HANKEL OPERATOR ON GENERAL HILBERT SPACES

  • Jyoti Bhola;Bhawna Gupta
    • 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 kth-order (C, r)-Hankel operators and kth-order (R, r)-Hankel operators (k ≥ 2) which are closely related to r-Hankel operators in such a way that a kth-order (C, r)-Hankel matrix is formed from rk-Hankel matrix on deleting every consecutive (k - 1) columns after the first column and a kth-order (R, rk)-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.

TRUNCATED HANKEL OPERATORS AND THEIR MATRICES

  • Lanucha, Bartosz;Michalska, Malgorzata
    • 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.

SLANT H-TOEPLITZ OPERATORS ON THE HARDY SPACE

  • Gupta, Anuradha;Singh, Shivam Kumar
    • 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.

COMPUTATION OF HANKEL MATRICES IN TERMS OF CLASSICAL KERNEL FUNCTIONS IN POTENTIAL THEORY

  • Chung, Young-Bok
    • Journal of the Korean Mathematical Society
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    • v.57 no.4
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    • pp.973-986
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    • 2020
  • In this paper, we compute the Hankel matrix representation of the Hankel operator on the Hardy space of a general bounded domain with respect to special orthonormal bases for the Hardy space and its orthogonal complement. Moreover we obtain the compact form of the Hankel matrix for the unit disc case with respect to these bases. One can see that the Hankel matrix generated by this computation turns out to be a generalization of the case of the unit disc from the single simply connected domain to multiply connected domains with much diversities of bases.

AN ELEMENTARY COMPUTATION OF HANKEL MATRICES ON THE UNIT DISC

  • Chung, Young-Bok
    • Honam Mathematical Journal
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    • v.43 no.4
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    • pp.691-700
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    • 2021
  • In this paper, we compute directly the Hankel matrix representation of the Hankel operator on the Hardy space of the unit disc without using any classical kernel functions with respect to special orthonormal bases for the Hardy space and its orthogonal complement. This gives an elementary proof for the formula.

TOEPLITZ AND HANKEL OPERATORS WITH CARLESON MEASURE SYMBOLS

  • Park, Jaehui
    • 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.

A study on the Hankel approximation of input delay systems (입력 시간지연 시스템의 한켈 근사화에 관한 연구)

  • Hwang, Lee-Cheol;Ha, Hui-Gwon;Lee, Man-Hyeong
    • Journal of Institute of Control, Robotics and Systems
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    • v.4 no.3
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    • pp.308-314
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    • 1998
  • This paper studies the problem of computing the Hankel singular values and vectors in the input delay systems. It is shown that the Hankel singular values are solutions to a transcendental equation and the Hankel singular vectors are obtained from the kernel of the matrix. The computation is carried out in state space framework. Finally, Hankel approximation of a simple example shows the usefulness of this study.

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PROPERTIES OF kth-ORDER (SLANT TOEPLITZ + SLANT HANKEL) OPERATORS ON H2(𝕋)

  • Gupta, Anuradha;Gupta, Bhawna
    • 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 kth-order slant Toeplitz operator with symbol 𝜙 and $S^k_{H_{\xi}}$ is a kth-order slant Hankel operator with symbol ξ. The spectral properties of operators Sk𝓜(𝜙,𝜙) (or simply Sk𝓜(𝜙)) are investigated on H2(𝕋). More precisely, it is proved that for k = 2, the Coburn's type theorem holds for Sk𝓜(𝜙). The conditions under which operators Sk𝓜(𝜙) commute are also explored.

LITTLE HANKEL OPERATORS ON WEIGHTED BLOCH SPACES IN Cn

  • Choi, Ki-Seong
    • Communications of the Korean Mathematical Society
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    • v.18 no.3
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    • pp.469-479
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    • 2003
  • Let B be the open unit ball in $C^{n}$ and ${\mu}_{q}$(q > -1) the Lebesgue measure such that ${\mu}_{q}$(B) = 1. Let ${L_{a,q}}^2$ be the subspace of ${L^2(B,D{\mu}_q)$ consisting of analytic functions, and let $\overline{{L_{a,q}}^2}$ be the subspace of ${L^2(B,D{\mu}_q)$) consisting of conjugate analytic functions. Let $\bar{P}$ be the orthogonal projection from ${L^2(B,D{\mu}_q)$ into $\overline{{L_{a,q}}^2}$. The little Hankel operator ${h_{\varphi}}^{q}\;:\;{L_{a,q}}^2\;{\rightarrow}\;{\overline}{{L_{a,q}}^2}$ is defined by ${h_{\varphi}}^{q}(\cdot)\;=\;{\bar{P}}({\varphi}{\cdot})$. In this paper, we will find the necessary and sufficient condition that the little Hankel operator ${h_{\varphi}}^{q}$ is bounded(or compact).