Compact operators with BMO symbols on multiply-connected domains

In this paper we study the problem of the boundedness and compactness of the Toeplitz operator T-phi on L-alpha(2)(Omega), where Omega is a multiply-connected domain and phi is not bounded. We find a necessary and sufficient condition when the symbol is BMO. For this class we also show that the vanishing at the boundary of the Berezin transform is a necessary and sufficient condition for compactness. The same characterization is shown to hold when we analyze operators which are finite sums of finite products of Toeplitz operators with unbounded symbols.