Optimal approximation of linear operators: a singular value decomposition approach

The purpose of this paper is to propose a definition of a set of singular values and a singular value decomposition associated with a linear operator defined on arbitrary normed linear spaces. This generalizes the usual notion of singular values and singular value decompositions to operators defined on spaces equipped with the p-norm, where p is arbitrary. Basic properties of these generalized singular values are derived and the problem of optimal rank approximation of linear operators is investigated in this context. We give sufficient conditions for the existence of optimal rank approximants in the p-induced norm and discuss an application of generalized singular values for the identification of dynamical systems from data