In this paper we study classical spaces of analytic functions which are convex cones and closed under the involution f -> 1/f. These include the spaces of positive real, positive real odd, and strictly positive real functions. These spaces are associated in the engineering literature with energy dissipation in the sense of Lyapunov. We discuss the geometric aspects of these spaces, in analogy with similar spaces in matrix theory which are also related to the matrix Lyapunov equation, stability and in view of a general theory of convex invertible cones. (c) 2007 Elsevier Inc. All rights reserved.42541700797813