• Title/Summary/Keyword: Toeplitz operators

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COMMUTATIVITY AND HYPONORMALITY OF TOEPLITZ OPERATORS ON THE WEIGHTED BERGMAN SPACE

  • Lu, Yufeng;Liu, Chaomei
    • Journal of the Korean Mathematical Society
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    • v.46 no.3
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    • pp.621-642
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    • 2009
  • In this paper we give necessary and sufficient conditions that two Toeplitz operators with monomial symbols acting on the weighted Bergman space commute. We also present necessary and sufficient conditions for the hyponormality of Toeplitz operators with some special symbols on the weighted Bergman space. All the results are stated in terms of the Mellin transform of the symbol.

TOEPLITZ OPERATORS ON HARDY AND BERGMAN SPACES OVER BOUNDED DOMAINS IN THE PLANE

  • Chung, Young-Bok;Na, Heui-Geong
    • Honam Mathematical Journal
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    • v.39 no.2
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    • pp.143-159
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    • 2017
  • In this paper, we show that algebraic properties of Toeplitz operators on both Bergman spaces and Hardy spaces on the unit disc essentially generalizes on arbitrary bounded domains in the plane. In particular, we obtain results for the uniqueness property and commuting problems of the Toeplitz operators on the Hardy and the Bergman spaces associated to bounded domains.

HYPONORMAL TOEPLITZ OPERATORS ON THE BERGMAN SPACE. II.

  • Hwang, In-Sung;Lee, Jong-Rak
    • Bulletin of the Korean Mathematical Society
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    • v.44 no.3
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    • pp.517-522
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    • 2007
  • In this paper we consider the hyponormality of Toeplitz operators $T_\varphi$ on the Bergman space $L_\alpha^2(\mathbb{D})$ with symbol in the case of function $f+\bar{g}$ with polynomials f and g. We present some necessary conditions for the hyponormality of $T_\varphi$ under certain assumptions about the coefficients of $\varphi$.

COMPACT TOEPLITZ OPERATORS

  • Kang, Si Ho
    • Honam Mathematical Journal
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    • v.35 no.3
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    • pp.343-350
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    • 2013
  • In this paper we prove that if Toeplitz operators $T^{\alpha}_u$ with symbols in RW satisfy ${\parallel}uk^{\alpha}_z{\parallel}_{s,{\alpha}{\rightarrow}0$ as $z{\rightarrow}{\partial}\mathbb{D}$ then $T^{\alpha}_u$ is compact and also prove that if $T^{\alpha}_u$ is compact then the Berezin transform of $T^{\alpha}_u$ equals to zero on ${\partial}\mathbb{D}$.

HYPONORMALITY OF TOEPLITZ OPERATORS ON THE BERGMAN SPACE

  • Hwang, In-Sung
    • Journal of the Korean Mathematical Society
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    • v.45 no.4
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    • pp.1027-1041
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    • 2008
  • In this paper we consider the hyponormality of Toeplitz operators $T_{\varphi}$ on the Bergman space $L_a^2{(\mathbb{D})$ in the cases, where ${\varphi}\;:=f+\bar{g}$ (f and g are polynomials). We present some necessary or sufficient conditions for the hyponormality of $T_{\varphi}$ under certain assumptions about the coefficients of ${\varphi}$.

HYPONORMALITY OF TOEPLITZ OPERATORS ON THE BERGMAN SPACE

  • Lee, Jongrak
    • Korean Journal of Mathematics
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    • v.15 no.2
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    • pp.185-193
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    • 2007
  • In this paper we consider the hyponormality of Toeplitz operators $T_{\varphi}$ on the Bergman space $L^2_a({\mathbb{D})$ with symbol in the case of function $f+{\overline{g}}$ with polynomials $f$ and $g$. We present some necessary conditions for the hyponormality of $T_{\varphi}$ under certain assumptions about the coefficients of ${\varphi}$.

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HYPONORMAL TOEPLITZ OPERATORS ON THE BERGMAN SPACE

  • Lee, Jong-Rak;Lee, You-Ho
    • Honam Mathematical Journal
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    • v.30 no.1
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    • pp.127-135
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    • 2008
  • In this paper we consider the hyponormality of Toeplitz operators $T_{\varphi}$ on the Bergman space $L^2_a(\mathbb{D})$ with symbol in the case of function f + $\overline{g}$ with polynomials f and g. We present some necessary conditions for the hyponormality of $T_{\varphi}$, under certain assumptions about the coefficients of ${\varphi}$.

ON HYPONORMALITY OF TOEPLITZ OPERATORS WITH POLYNOMIAL AND SYMMETRIC TYPE SYMBOLS

  • Hazarika, Munmun;Phukon, Ambeswar
    • Bulletin of the Korean Mathematical Society
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    • v.48 no.3
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    • pp.617-625
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    • 2011
  • In [6], it was shown that hyponormality for Toeplitz operators with polynomial symbols can be reduced to classical Schur's algorithm in function theory. In [6], Zhu has also given the explicit values of the Schur's functions ${\Phi}_0$, ${\Phi}_1$ and ${\Phi}_2$. Here we explicitly evaluate the Schur's function ${\Phi}_3$. Using this value we find necessary and sufficient conditions under which the Toeplitz operator $T_{\varphi}$ is hyponormal, where ${\varphi}$ is a trigonometric polynomial given by ${\varphi}(z)$ = ${\sum}^N_{n=-N}a_nz_n(N{\geq}4)$ and satisfies the condition $\bar{a}_N\(\array{a_{-1}\\a_{-2}\\a_{-4}\\{\vdots}\\a_{-N}}\)=a_{-N}\;\(\array{\bar{a}_1\\\bar{a}_2\\\bar{a}_4\\{\vdots}\\\bar{a}_N}\)$. Finally we illustrate the easy applicability of the derived results with a few examples.