In this paper, we analyze l(2)-stability of infinite dimensional discrete autonomous systems given in a state space form with state transition matrix being a Laurent polynomial matrix A(sigma, sigma(-1)) in the shift operator a. We give sufficient conditions and necessary conditions for l(2)-stability of such systems. We then use the theory of l(2)-stability, thus developed, to analyze l(2)-stability of discrete 2-D autonomous systems. We achieve this by showing how a discrete 2-D autonomous system can be converted to an equivalent infinite dimensional state space discrete autonomous system, where the state transition matrix turns out to be a Laurent polynomial matrix in the shift operator. Finally, we provide some easy-to-check numerical tests for l(2)-stability of the above-mentioned type of systems. (C) 2017 Elsevier Ltd. All rights reserved.