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Positive operators on the Bergman space and Berezin transform


Abstract

Let $\mathbb{D}=\{{z\in\mathbb{C}:|z|<1}\}$ and $L^2_a(\mathbb{D})$ be the Bergman space of the disk. In thispaper we characterize the class $\mathcal{A}\subset L^\infty(\mathbb{D})$ such that if $\phi,\psi\in\mathcal{A},\alpha\geq 0$ and $0\leq\phi\leq\alpha\psi$ then there exist positive operators $S,T\in\mathcal{L}(L^2_a(\mathbb{D}))$ such that $\phi(z)=\widetilde{S}(z)\leq\alpha\widetilde{T}(z)=\alpha\psi(z)$ for all $z\in\mathbb{D}$. Further, we have shown that if $S$ and $T$ are two positive operators in $\mathcal{L}(L^2_a(\mathbb{D}))$ and $T$ is invertible then there exists a constant $\alpha\geq0$ such that $\widetilde{S}(z)\leq\alpha\widetilde{T}(z)$ for all $z\in\mathbb{D}$ and $\widetilde{S},\widetilde{T}\in\mathcal{A}$. Here $\mathcal{L}(L^2_a(\mathbb{D}))$ is the space of all boundedlinear operators from $L^2_a(\mathbb{D})$ into $L^2_a(\mathbb{D})$ and $\widetilde{A}(z)=\langle Ak_z,k_z\rangle$ is the Berezintrans form of $A\in\mathcal{L}(L^2_a(\mathbb{D}))$ and $k_z$ is thenormalized reproducing kernel of $L^2_a(\mathbb{D})$. Applications of these results are also obtained.


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Article Information

TitlePositive operators on the Bergman space and Berezin transform
SourceMethods Funct. Anal. Topology, Vol. 17 (2011), no. 3, 204-210
MathSciNet MR2857723
CopyrightThe Author(s) 2011 (CC BY-SA)

Authors Information

Namita Das
Department of Mathematics, Utkal University, Vani Vihar, Bhubaneswar-751004, Orissa, India

Madhusmita Sahoo
School of Applied Sciences (Mathematics), KIIT University, Campus-3 (Kathajori Campus) Bhubaneswar-751024, Orissa, India 


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Citation Example

Namita Das and Madhusmita Sahoo, Positive operators on the Bergman space and Berezin transform, Methods Funct. Anal. Topology 17 (2011), no. 3, 204-210.


BibTex

@article {MFAT580,
    AUTHOR = {Das, Namita and Sahoo, Madhusmita},
     TITLE = {Positive operators on the Bergman space and Berezin transform},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {17},
      YEAR = {2011},
    NUMBER = {3},
     PAGES = {204-210},
      ISSN = {1029-3531},
       URL = {http://mfat.imath.kiev.ua/article/?id=580},
}


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