Abstract
Let T f be a Toeplitz operator on the Segal–Bargmann space or the standard weighted Bergman space over a bounded symmetric domain \({\Omega \subset {\bf C}^n}\) 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 C n and the Bergman metric on \({\Omega}\), 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 \({\Omega.}\) This observation substantially extends a result in Coburn (Indiana Univ. Math. J. 23:433–439, 1973).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bauer W.: Integr. Equ. Oper. Theory 52, 1–15 (2005)
Bauer W., Coburn L.A., Isralowitz J.: Heat flow, BMO, and the compactness of Toeplitz operators. J. Funct. Anal. 259, 57–78 (2010)
Bauer, W., Coburn, L.A.: Heat flow, weighted Bergman spaces and real analytic Lipschitz approximation. J. reine angew. Math. (doi:10.1515/crelle-2015-0016) (to appear in)
Békollé D., Berger C., Coburn L.A., Zhu K.H.: BMO in the Bergman metric on bounded symmetric domains. J. Funct. Anal. 93, 310–350 (1990)
Berger C., Coburn L.A.: Heat flow and Berezin–Toeplitz estimates. Am. J. Math. 116(3), 563–590 (1994)
Cartan E.: Sur les domaines bornés homogènes de l’ espace de n-variables complexes. Abh. Math. Semin. Univ. Hamburg 11, 116–162 (1935)
Coburn L.A.: Sharp Berezin Lipschitz estimates. Proc. Am. Math. Soc. 135, 1163–1168 (2007)
Coburn L.A.: Singular integral operators and Toeplitz operators on odd spheres. Indiana Univ. Math. J. 23, 433–439 (1973)
Engliš M.: Compact Toeplitz operators via the Berezin transform on bounded symmetric domains. Integr. Equ. Oper. Theory 33, 426–455 (1999)
Faraut J., Koranyi A.: Function spaces and reproducing kernels on bounded symmetric domains. J. Funct. Anal. 88, 64–89 (1990)
Helgason, S.: Differential geometry, Lie groups, and symmetric spaces. Graduate Studies in Mathematics 34, AMS Providence, Rhode Island (2001)
Issa H.: Compact Toeplitz operators for weighted Bergman spaces on bounded symmetric domains. Integr. Equ. Oper. Theory 70, 569–582 (2011)
Koecher M.: An elementary approach to bounded symmetric domains. Rice Univ. Press, Houston (1969)
Loos, O.: Bounded symmetric domains and Jordan pairs. Mathematical Lectures. University of California at Irvine, Irvine (1977)
Upmeier, H.: Toeplitz Operators and Index Theory in Several Complex Variables. Operator Theory: Advances and Applications 81, Birkhäuser (1996)
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by an “Emmy-Noether scholarship” (BA 3793/1-1) of the DFG (Deutsche Forschungsgemeinschaft).
Rights and permissions
About this article
Cite this article
Bauer, W., Coburn, L.A. Toeplitz Operators with Uniformly Continuous Symbols. Integr. Equ. Oper. Theory 83, 25–34 (2015). https://doi.org/10.1007/s00020-015-2235-4
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00020-015-2235-4