Toeplitz Operators with Uniformly Continuous Symbols
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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).
Keywords
Segal–Bargmann space heat transform bounded symmetric domain Bergman metricMathematics Subject Classification
32M15 32A36 47B35Preview
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References
- 1.Bauer W.: Integr. Equ. Oper. Theory 52, 1–15 (2005)MATHCrossRefGoogle Scholar
- 2.Bauer W., Coburn L.A., Isralowitz J.: Heat flow, BMO, and the compactness of Toeplitz operators. J. Funct. Anal. 259, 57–78 (2010)MATHMathSciNetCrossRefGoogle Scholar
- 3.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)
- 4.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)MATHMathSciNetCrossRefGoogle Scholar
- 5.Berger C., Coburn L.A.: Heat flow and Berezin–Toeplitz estimates. Am. J. Math. 116(3), 563–590 (1994)MATHMathSciNetCrossRefGoogle Scholar
- 6.Cartan E.: Sur les domaines bornés homogènes de l’ espace de n-variables complexes. Abh. Math. Semin. Univ. Hamburg 11, 116–162 (1935)MathSciNetCrossRefGoogle Scholar
- 7.Coburn L.A.: Sharp Berezin Lipschitz estimates. Proc. Am. Math. Soc. 135, 1163–1168 (2007)MATHMathSciNetCrossRefGoogle Scholar
- 8.Coburn L.A.: Singular integral operators and Toeplitz operators on odd spheres. Indiana Univ. Math. J. 23, 433–439 (1973)MATHMathSciNetCrossRefGoogle Scholar
- 9.Engliš M.: Compact Toeplitz operators via the Berezin transform on bounded symmetric domains. Integr. Equ. Oper. Theory 33, 426–455 (1999)MATHCrossRefGoogle Scholar
- 10.Faraut J., Koranyi A.: Function spaces and reproducing kernels on bounded symmetric domains. J. Funct. Anal. 88, 64–89 (1990)MATHMathSciNetCrossRefGoogle Scholar
- 11.Helgason, S.: Differential geometry, Lie groups, and symmetric spaces. Graduate Studies in Mathematics 34, AMS Providence, Rhode Island (2001)Google Scholar
- 12.Issa H.: Compact Toeplitz operators for weighted Bergman spaces on bounded symmetric domains. Integr. Equ. Oper. Theory 70, 569–582 (2011)MATHMathSciNetCrossRefGoogle Scholar
- 13.Koecher M.: An elementary approach to bounded symmetric domains. Rice Univ. Press, Houston (1969)MATHGoogle Scholar
- 14.Loos, O.: Bounded symmetric domains and Jordan pairs. Mathematical Lectures. University of California at Irvine, Irvine (1977)Google Scholar
- 15.Upmeier, H.: Toeplitz Operators and Index Theory in Several Complex Variables. Operator Theory: Advances and Applications 81, Birkhäuser (1996)Google Scholar