Toeplitz Operators on Arveson and Dirichlet Spaces

We define Toeplitz operators on all Dirichlet spaces on the unit ball of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb{C}^{N}$$ \end{document} and develop their basic properties. We characterize bounded, compact, and Schatten-class Toeplitz operators with positive symbols in terms of Carleson measures and Berezin transforms. Our results naturally extend those known for weighted Bergman spaces, a special case applies to the Arveson space, and we recover the classical Hardy-space Toeplitz operators in a limiting case; thus we unify the theory of Toeplitz operators on all these spaces. We apply our operators to a characterization of bounded, compact, and Schatten-class weighted composition operators on weighted Bergman spaces of the ball. We lastly investigate some connections between Toeplitz and shift operators.