Toeplitz Operators with Uniformly Continuous Symbols

Let T_f be a Toeplitz operator on the Segal–Bargmann space or the standard weighted Bergman space over a bounded symmetric domain Ω⊂Cⁿ with possibly unbounded symbol f. Combining recent results in Bauer et al. (J. Funct. Anal. 259:57–78, 2010), Bauer et al. (J. reine angew. Math. doi:10.1515/crelle-2015-0016), Issa (Integr. Equ. Oper. Theory 70:569–582, 2011) we show that in the case of uniformly continuous symbols f with respect to the Euclidean metric on Cⁿ and the Bergman metric on Ω, respectively, the operator T_f is bounded if and only if f is bounded. Moreover, T_f is compact if and only if f vanishes at the boundary of Ω. This observation substantially extends a result in Coburn (Indiana Univ. Math. J. 23:433–439, 1973).