Local objects in the theory of Toeplitz operators (Q1086812)

R. Douglas associated with every Toeplitz operator a family of local Toeplitz opertors to study the Fredholm theory of Toeplitz operators. In the present paper local objects are introduced which can be used to study approximation of block Toeplitz operators by their finite sections or by their harmonic extension. So it turns out that there is a unified approach to quite different problems concerning Toeplitz operators, which, undoubtedly, provides new insights into the common features but also into the significant differences between various ''Toeplitz-like results''. The method developed in this paper yields new results on Fredholmness, harmonic approximation, and finite sections of block Toeplitz operators (involving, in particular, a new first Szegö limit theorem) and offers a possibility of computing the index of block Toeplitz operators generated by matrix functions that are locally sectorial over QC.