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Compact operators and Toeplitz algebras on multiply-connected domains |
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| Authors: | Mirjana Jovovic Dechao Zheng |
| Affiliation: | a University of Hawai‘i at Mānoa, Department of Mathematics, 2565 McCarthy Mall, Honolulu, HI 96822, United States b The Center of Mathematics, Chongqing University, Chongqing, 401331, PR China c Vanderbilt University, Department of Mathematics, 1326 Stevenson Center, Nashville, TN 37240, United States |
| Abstract: | If Ω is a smoothly bounded multiply-connected domain in the complex plane and S belongs to the Toeplitz algebra τ of the Bergman space of Ω, we show that S is compact if and only if its Berezin transform vanishes at the boundary of Ω. We also show that every element S in T, the C?-subalgebra of τ generated by Toeplitz operators with symbols in H∞(Ω), has a canonical decomposition  for some R in the commutator ideal CT; and S is in CT iff the Berezin transform  vanishes identically on the set M1 of trivial Gleason parts. |
| Keywords: | Toeplitz algebra Berezin transform Bergman space Commutator ideal |
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