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Numerical range and Berezin range of weighted composition operators on weighted Dirichlet spaces

Somdatta Barik, Anirban Sen, Kallol paul

TL;DR

The paper investigates numerical and Berezin ranges for weighted composition operators on weighted Dirichlet spaces $D_s$ with $0<s<1$. It derives precise criteria for when the origin lies in the numerical range, and establishes when $W(C_{\psi,\phi};D_s)$ contains circular or elliptical discs, including explicit radii and axis lengths in terms of the symbols and coefficients. It then defines Weyl-type weighted composition operators and determines their Berezin range and ber, including exact values in key cases and a reverse power inequality for the Berezin radius. Finally, it analyzes the convexity of Berezin ranges for canonical symbol families, yielding sharp conditions for both linear and Blaschke-type maps. These results deepen the understanding of operator geometry on $D_s$ and have implications for spectral and numerical-range studies of weighted composition operators.

Abstract

We investigate the numerical ranges of weighted composition operators on weighted Dirichlet spaces, focusing on the properties of the inducing functions. We identify conditions on these functions under which the origin lies in the interior of the numerical range. The geometric structure of the numerical range is also analyzed, determining when it contains a circular or elliptical disc and computing the corresponding radius. Next, we introduce a class of Weyl-type weighted composition operators and obtain their Berezin range and Berezin number. Finally, we characterize the convexity of the Berezin range for weighted composition operators on these spaces.

Numerical range and Berezin range of weighted composition operators on weighted Dirichlet spaces

TL;DR

The paper investigates numerical and Berezin ranges for weighted composition operators on weighted Dirichlet spaces with . It derives precise criteria for when the origin lies in the numerical range, and establishes when contains circular or elliptical discs, including explicit radii and axis lengths in terms of the symbols and coefficients. It then defines Weyl-type weighted composition operators and determines their Berezin range and ber, including exact values in key cases and a reverse power inequality for the Berezin radius. Finally, it analyzes the convexity of Berezin ranges for canonical symbol families, yielding sharp conditions for both linear and Blaschke-type maps. These results deepen the understanding of operator geometry on and have implications for spectral and numerical-range studies of weighted composition operators.

Abstract

We investigate the numerical ranges of weighted composition operators on weighted Dirichlet spaces, focusing on the properties of the inducing functions. We identify conditions on these functions under which the origin lies in the interior of the numerical range. The geometric structure of the numerical range is also analyzed, determining when it contains a circular or elliptical disc and computing the corresponding radius. Next, we introduce a class of Weyl-type weighted composition operators and obtain their Berezin range and Berezin number. Finally, we characterize the convexity of the Berezin range for weighted composition operators on these spaces.
Paper Structure (5 sections, 19 theorems, 71 equations)

This paper contains 5 sections, 19 theorems, 71 equations.

Table of Contents

  1. Introduction and preliminaries
  2. Containment of zero in $W(C_{\psi,\phi}; \mathcal{D}_s)$
  3. Containment of circle or ellipse in $W(C_{\psi,\phi}; \mathcal{D}_s)$
  4. Weyl-type weighted composition operators on $\mathcal{D}_s$
  5. Convexity of Berezin range of composition operators on $\mathcal{D}_s$

Key Result

Theorem 1.1

Suppose $\phi$ is a holomorphic self-map of $\mathbb{D}$ and $\psi\in\mathcal{H}(\mathbb{D})$ with $C_{\psi,\phi}\in\mathcal{B}(\mathcal{D}_{s})$. $(i)$ If $\phi$ is identity map on $\mathbb D$ and $\psi$ has a zero in $\mathbb{D}$ then $0\in W(C_{\psi,\phi}; \mathcal{D}_s).$$(ii)$ If $\phi$ is ide

Theorems & Definitions (39)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 29 more