Discrete one-dimensional zero-order pseudodifferential operators on spaces with Muckenhoupt weight

This paper explores operators on $\ell^p$ over the integers with an arbitrary Muckenhoupt weight. The main result is a Fredholm criterion for the operators in the Banach algebra generated by zero-order pseudodifferential operators with piecewise continuous symbols. In contrast to the common power weights, general Muckenhoupt weights may produce massive parts in the essential spectrum of these operators.