Toeplitz operators on large vector-valued Fock spaces

TL;DR

This paper extends the scalar theory of Toeplitz operators on Fock spaces to large vector-valued Fock spaces with Dall'Ara weights, establishing a comprehensive framework for boundedness, compactness, and Schatten-class membership. The authors introduce and utilize both scalar and operator-valued Berezin transforms, together with averaging operators and Carleson-type conditions, to characterize $T_G$ on $F^2_\phi(\mathcal{H})$. They develop vector-valued reproducing kernel theory, derive precise kernel and projection estimates, and deploy $r$-lattice decompositions to obtain equivalences that unify the vector-valued and scalar settings. The resulting criteria cover boundedness, compactness, and $p$-Schatten class membership, providing a robust toolkit for operator theory on vector-valued Fock spaces with general doubling-type weights and enabling further applications in several complex variables and functional analysis.

Abstract

We characterize boundedness and compactness of Toeplitz operators on large vector-valued Fock spaces with Dall'Ara's weights [Adv.\ Math., 285 (2015) 1706--1740] in terms of generalized Berezin transforms, averaging functions, and Carleson measures. To determine Schatten class Toeplitz operators, we introduce the operator-valued Berezin transform and averaging functions.

Theorems & Definitions (31)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Definition 2.1
• Lemma 2.2: See WeightedCompositionHicham, Lemma A
• Lemma 2.3: See arroussi2022toeplitz, Lemma 2.3
• Lemma 2.4
• proof
• Remark 2.5