Toeplitz operators on large vector-valued Fock spaces

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.