Numerical range of Toeplitz and Composition operators on weighted Bergman spaces
Anirban Sen, Subhadip Halder, Riddhick Birbonshi, Kallol Paul
TL;DR
This work analyzes the numerical ranges of Toeplitz operators with harmonic symbols and weighted composition operators on weighted Bergman spaces $L_a^2(dA_{\alpha})$. It establishes a complete description of $W(T_{\phi})$ when $\phi$ is harmonic and derives precise shapes for $W(C_{\psi,\phi})$, including when the range contains a circle or an ellipse, by leveraging reproducing kernels, $\alpha$-essential ranges, and a decomposition into invariant subspaces $L_j$+. The results show that harmonicity is essential for the Toeplitz characterization and provide concrete criteria and formulas for zero-inclusion, convex hulls, and circular/elliptical containments, along with explicit radii and axes. These findings advance the operator-theoretic understanding of Toeplitz and weighted composition operators on weighted Bergman spaces and offer tools for estimating numerical radii and related spectral quantities.
Abstract
In this paper we completely describe the numerical range of Toeplitz operators on weighted Bergman spaces with harmonic symbol. We also characterize the numerical range of weighted composition operators on weighted Bergman spaces and classify some sets which are the numerical range of composition operators. We investigate the inclusion of zero in the numerical range, and compute the radius of circle and ellipse contained in the numerical range of weighted composition operators on weighted Bergman spaces.