The Factorization and Representation of Operators in the Algebra Generated by Toeplitz Operators

In this paper, we study the factorization and the representation of Fredholm operators belonging to the algebra $\mathcal{R}$ generated by inversion and composition of Toeplitz integral operators. The operators in $\mathcal{R}$ 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, we derive a set of efficient algorithms (generalized fast-Cholesky and Levinson recursions) 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.We also obtain several important representation theorems for the decomposition of operators in $\mathcal{R}$ in terms of sums of products of lower times upper triangular Toeplitz operators. These results can be used to approximate operators corresponding to noncausa...