| Popis: |
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 S = T S ˜ + R for some R in the commutator ideal C T ; and S is in C T iff the Berezin transform S ˜ vanishes identically on the set M 1 of trivial Gleason parts. |
