Toeplitz Operators on Fock Space $F_α^{\infty}$

TL;DR

The paper studies when positive measures are $(\infty,q)$ or $(p,\infty)$ (vanishing) Fock-Carleson measures via their Berezin transforms and analyzes Toeplitz operators on the maximal Fock space $F_{\alpha}^{\infty}$. It establishes Berezin-transform-based characterizations of Carleson-type properties and derives precise boundedness/compactness criteria for $T_{\mu}$ and $T_{\varphi}$ (with $\varphi\in BMO$) in terms of $t$-Berezin transforms $\tilde{\mu}_{t}$ and $\tilde{\varphi}_{t}$. The results show equivalences between $(\infty,q)$ and $(p,q)$ Carleson measures, as well as between $(p,\infty)$ and $(p,q)$ notions for finite mass, and they extend to symbols in $BMO$, clarifying how Berezin transforms govern operator behavior on $F_{\alpha}^{\infty}$. These findings enhance the toolbox for operator theory on generalized Fock spaces by providing tangible, transform-based criteria.

Abstract

In this paper, we study necessary and sufficient conditions for a positive Borel measure $μ$ on the complex space $\mathbb{C}$ to be a $(\infty,q)$ or $(p,\infty)$ (vanishing) Fock-Carleson measure through its Berezin transform. Then we discuss boundedness and compactness of the Toeplitz operator $T_μ$ with a positive Borel measure $μ$ as symbol on Fock space $F_α^{\infty}$. Furthermore, we charaterize these properties of the Toeplitz operator T_{\varphi}$ with a symbol $\varphi$ which is in $BMO$.

Theorems & Definitions (32)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Lemma 2.1
• proof
• Theorem 2.1
• proof
• Theorem 2.2
• proof
• Theorem 2.3