Abstract.
We define Toeplitz operators on all Dirichlet spaces on the unit ball of \(\mathbb{C}^{N}\) 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.
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The research of the second author is partially supported by a Fulbright grant.
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Alpay, D., Kaptanoğlu, H.T. Toeplitz Operators on Arveson and Dirichlet Spaces. Integr. equ. oper. theory 58, 1–33 (2007). https://doi.org/10.1007/s00020-007-1493-1
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DOI: https://doi.org/10.1007/s00020-007-1493-1