FACTORIZATION AND REPRESENTATION OF OPERATORS IN THE ALGEBRA GENERATED BY TOEPLITZ OPERATORS

A study is made of factorization and the representation of Fredholm operators belonging to the algebra R generated by inversion and composition of Toeplitz integral operators. The operators in R have the interesting property of being close to Toeplitz (in a sense quantifiable by an integer-valued index alpha ) and, at the same time, of being dense in the space of arbitrary kernels. By using these properties, a set of efficient algorithms (generalized fast-Cholesky and Levinson recursions) is derived for the factorization and the inversion of arbitrary Fredholm operators. The computational burden of these algorithms depends on how close (as measured by the index alpha ) these operators are to being Toeplitz. Several important representation theorems for the decomposition of operators in R in terms of sums of products of lower times upper triangular Toeplitz operators are also obtained.