Exponential localization of linear response in networks with exponentially decaying coupling

Let S be a countable metric space with metric d, for each s is an element of S let X-s, Y-s be Banach spaces, and let X, Y be the subsets of x(s is an element of S), x(s is an element of S)Y(s), respectively, with finite supremum norm over their factors. Let L : X --> Y be an invertible 'exponentially local' bounded linear map, i.e. such that for some zeta > 1, GRAPHICS Let u is an element of Y be exponentially localized around a site o is an element of S. Then the response upsilon = L(-1)u is also exponentially localized about o. This linear result is of fundamental importance to a wide variety of nonlinear problems, including spatial localization of discrete breathers and bipolarons. For illustration, a simple application is given to equilibria of networks of bistable units. Finally, the result is generalized to maps between product spaces with arbitrary norms based on the norms on the factors.