Fredholmness and finite section method for Toeplitz operators in $l^p(Z_{+}\times Z_{+})$ with piecewise continuous symbols I

Abstract

We consider discrete Toeplitz operators on the space $l^p$ over the quarter-plane for a class of piecewise continuous symbols. This class of symbols is usually denoted by $P{C}_p({\mathbb{T}}^2)$ and it contains, in particular, all finite sums of the form $\Sigma a_i(\xi )b_i(\eta ),(\xi ,\eta )\in {\mathbb{T}}^2$, where $a_i$ and $b_i$ are of bounded variation. Necessary and sufficient conditions for Fredholmness of such operators and for the applicability of the finite section method to them are obtained. The present part I contains the necessary definitions, the formulation of the main results, and the proofs of the necessity of the given conditions. Their sufficiency will be proved in part II of this work.