A generalization of the Brown-Halmos theorems for the unit ball Article (Faculty180)

cited authors

  • Le, Trieu L; Tikaradze, Akaki


  • In this paper we generalize the classical theorems of Brown and Halmos about algebraic properties of Toeplitz operators to the Bergman space over the unit ball in several complex variables. A key result, which is of independent interest, is the characterization of summable functions $u$ on the unit ball whose Berezin transform $B(u)$ can be written as a finite sum $\sum_{j}f_j\,\bar{g}_j$ with all $f_j, g_j$ being holomorphic. In particular, we show that such a function must be pluriharmonic if it is sufficiently smooth and bounded. We also settle an open question about $\mathcal{M}$-harmonic functions. Our proofs employ techniques and results from function and operator theory as well as partial differential equations.

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