Guangfu Cao

This paper investigates the essential norm of Toeplitz operators $\mathcal{T}_μ$ acting from the Bergman space $A_ω^p$ to $A_ω^q$ ($1 < p \leq q < \infty$) on the unit ball, where $μ$ is a positive Borel measure and $ω\in \mathcal{D}$ (a class of doubling weights). Leveraging the geometric properties of Carleson blocks and the structure of radial doubling weights, we establish sharp estimates for the essential norm in terms of the asymptotic behavior of $μ$ near the boundary. As a consequence, we resolve the boundedness-to-compactness transition for these operators when $1 < q < p<\infty$, showing that the essential norm vanishes exactly. These results generalize classical theorems for the unweighted Bergman space ($ω\equiv 1$) and provide a unified framework for studying Toeplitz operators under both radial and non-radial doubling weights in higher-dimensional settings.

Let $B(\Omega )$ be a Banach space of holomorphic functions on a bounded connected domain $\Omega $ in ${{\mathbb C}^n}$ . In this paper, we establish a criterion for $B(\Omega )$ to be reflexive via evaluation functions on $B(\Omega )$ , that is, $B(\Omega )$ is reflexive if and only if the evaluation functions span the dual space $(B(\Omega ))^{*} $ .

Let $B(\Omega)$ be the Banach space of holomorphic functions on a bounded connected domain $\Omega$ in $\mathbb C^n$, which contains the ring of polynomials on $\Omega $. In this paper, we first establish a criterion for $B(\Omega )$ to be reflexive via evaluation functions on $B(\Omega )$, that is, $B(\Omega )$ is reflexive if and only if the evaluation functions span the dual spaces $(B(\Omega ))^{*} $. Moreover, under suitable assumptions on $\Omega$ and $B(\Omega)$, we establish a characterization of the composition operator $C_\varphi$ to be a Fredholm operator on $B(\Omega)$ via the property of the holomorphic self-map $\varphi:\Omega\to\Omega$. Our new approach utilizes the symbols of composition operators to construct a linearly independent function sequence, which bypasses the use of boundary behavior of reproducing kernels as those may not be applicable in our general setting.