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Some results for weighted Bergman space operators via Berezin symbols

  • Ramiz Tapdigoglu
Published/Copyright: May 24, 2024
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 74 Issue 2

Abstract

We consider the weighted Bergman space Aα2(𝔻) of analytic functions f on the unit disc 𝔻 = {z ∈ ℂ : |z| < 1} such that

f A α 2 2 := ∫ D f z 2 d A α z < + ∞ ,

where d Aα(z) = (α+1) (1 − |z|2)α d A(z) and dAz=dxdyπ (the normalized Lebesgue area measure). We investigate the generalized Riccati operator equation

X A 1 X + A 2 X A 3 + A 4 Y A 5 + A 6 = 0

with Ai ∈ 𝓑(Aα2(𝔻) ), i = 1, 6. Also we study the solvability of the operator equation

X 1 T φ 1 + X 2 T φ 2 + ⋯ + X n T φ n = I A α 2 ,

in the set of Toeplitz operators on the weighted Bergman space Aα2(𝔻). Moreover, we characterize compactness of the operator Tφ Tψ − Tη in terms of Berezin transforms.

Keywords: Weighted Bergman space; Berezin transform; Riccati equation; Toeplitz operator
MSC 2010: 47B35

Acknowledgement

The author thanks to the anonymous referees for their helpful comments that improved the quality of the manuscript.

  1. Communicated by Marek Balcerzak

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Received: 2023-02-11
Accepted: 2023-06-04
Published Online: 2024-05-24
Published in Print: 2024-04-25

© 2024 Mathematical Institute Slovak Academy of Sciences

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  2. Topological representation of some lattices
  3. Exact-m-majority terms
  4. Polynomials whose coefficients are generalized Leonardo numbers
  5. A study on error bounds for Newton-type inequalities in conformable fractional integrals
  6. Improved conditions for the distributivity of the product for σ-algebras with respect to the intersection
  7. Close-to-convex functions associated with a rational function
  8. Complete monotonicity for a ratio of finitely many gamma functions
  9. Class of bounds of the generalized Volterra functions
  10. Some new uniqueness and Ulam–Hyers type stability results for nonlinear fractional neutral hybrid differential equations with time-varying lags
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  14. Weighted 1MP and MP1 inverses for operators
  15. Non-commutative effect algebras, L-algebras, and local duality
  16. Operator Bohr-type inequalities
  17. Some results for weighted Bergman space operators via Berezin symbols
  18. Lower separation axioms in bitopogenous spaces
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  20. Irreducibility of strong size levels
https://doi.org/10.1515/ms-2024-0036
Keywords for this article
Weighted Bergman space; Berezin transform; Riccati equation; Toeplitz operator

Articles in the same Issue

  1. Right algebras in Sup and the topological representation of semi-unital and semi-integral quantales, revisited
  2. Topological representation of some lattices
  3. Exact-m-majority terms
  4. Polynomials whose coefficients are generalized Leonardo numbers
  5. A study on error bounds for Newton-type inequalities in conformable fractional integrals
  6. Improved conditions for the distributivity of the product for σ-algebras with respect to the intersection
  7. Close-to-convex functions associated with a rational function
  8. Complete monotonicity for a ratio of finitely many gamma functions
  9. Class of bounds of the generalized Volterra functions
  10. Some new uniqueness and Ulam–Hyers type stability results for nonlinear fractional neutral hybrid differential equations with time-varying lags
  11. Existence of positive solutions to a class of boundary value problems with derivative dependence on the half-line
  12. Solving Fredholm integro-differential equations involving integral condition: A new numerical method
  13. Bounds of some divergence measures on time scales via Abel–Gontscharoff interpolation
  14. Weighted 1MP and MP1 inverses for operators
  15. Non-commutative effect algebras, L-algebras, and local duality
  16. Operator Bohr-type inequalities
  17. Some results for weighted Bergman space operators via Berezin symbols
  18. Lower separation axioms in bitopogenous spaces
  19. Fisher information in order statistics and their concomitants for Cambanis bivariate distribution
  20. Irreducibility of strong size levels
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