Abstract
We characterize the multiplication operators with closed range on the Bergman space in terms of the Berezin transform, and apply this characterization to finite products of interpolating Blaschke products. We give some necessary and some sufficient conditions for invertibility of general Toeplitz operators on the Bergman space. We determine the Fredholm Toeplitz operators with \(BMO^1\) symbols and the invertible Toeplitz operators with nonnegative symbols, when their Berezin transform is bounded and of vanishing oscillation.
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Axler, S., Zheng, D.: Compact operators via the Berezin transform. Indiana Univ. Math. J. 47, 387–400 (1998)
Axler, S., Zheng, D.: The Berezin transform on the Toeplitz algebra. Studia Math. 127, 113–136 (1998)
Chicoń, K.: Closed range multiplication operators on weighted Bergman spaces. Nonlinear Anal. 60, 37–48 (2005)
Davidson, K., Douglas, R.: The generalized Berezin transform and commutator ideals. Pacific J. Math. 222, 29–56 (2005)
Ghatage, P., Tjani, M.: Closed range composition operators on Hilbert function spaces. J. Math. Anal. Appl. 431, 841–866 (2015)
Gorkin, P.: Functions not vanishing on trivial Gleason parts of Douglas algebras. Proc. Am. Math. Soc. 104, 1086–1090 (1998)
Luecking, D.: Inequalities on Bergman spaces. Illinois J. Math. 25, 1–11 (1981)
Luecking, D.: Characterization of certain classes of Hankel operators on the Bergman spaces. J. Funct. Anal. 110, 247–271 (1992)
Luecking, D.: Bounded composition operators with closed range on the Dirichlet space. Proc. Am. Math. Soc. 128, 1109–1116 (2000)
Marshal, D.E., Stray, A.: Interpolating Blaschke products. Pacific J. Math. 173, 491–499 (1996)
McDonald, G., Sundberg, C.: Toeplitz operators on the disc. Indiana Univ. Math. J. 28, 595–611 (1979)
Nikolskii, N.K.: Treatise on the Shift Operator. Springer, Berlin (1986)
Stroethoff, K., Zheng, D.: Toeplitz and Hankel operators on Bergman spaces. Trans. Am. Math. Soc. 329, 773–794 (1992)
Suarez, D.: The essential norm in the Toeplitz algebra on \(A^p (\mathbb{B}_n)\). Indiana Univ. Math. J. 56, 2185–2232 (2007)
Taskinen, J., Virtanen, J.: Toeplitz operators on Bergman spaces with locally integrable symbols. Rev. Math. Iberoamericana 26, 693–706 (2010)
Zhao, X., Zheng, D.: Invertibility of Toeplitz operators via Berezin Transforms. J. Oper. Theory 75, 475–495 (2016)
Zhu, K.: VMO, ESV, and Toeplitz operators on the Bergman space. Trans. Am. Math. Soc. 302, 617–646 (1987)
Zhu, K.: Operator Theory in Function Spaces. Marcel Dekker, New York (1990)
Zorboska, N.: Toeplitz operators with BMO symbols and the Berezin transform. Int. J. Math. Sci. 46, 2929–2945 (2003)
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Canadian Network for Research and Innovation in Machining Technology, Natural Sciences and Engineering Research Council of Canada (CA) (Grant No. 1304332).
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Communicated by Raymond Mortini.
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Zorboska, N. Closed Range Type Properties of Toeplitz Operators on the Bergman Space and the Berezin Transform. Complex Anal. Oper. Theory 13, 4027–4044 (2019). https://doi.org/10.1007/s11785-019-00949-4
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DOI: https://doi.org/10.1007/s11785-019-00949-4
Keywords
- Multiplication operator
- Toeplitz operator
- Bergman space
- Berezin transform
- Closed range operator
- Invertible operator
- Fredholm operator
- Interpolating Blaschke product