A general theorem on the limitation of power consumption

The characteristic values of the open loop impedance (admittance) operator of a linear passive reciprocal n-port impose certain natural bounds on the effective total energy consumptions in the network. It is shown that the characteristic values of a linear system and the energy dissipated by it are interrelated via arithmetic and geometric inequalities. The lien between the two concepts is the convexity of quadratic forms associated with non-negative operators, and the minimax property of their eigenvalues. The energy limitation for passive system is best exhibited by a natural setting of the problem in a function space format: a format lacking thus far in the classical network theory approach. In this manner, the interplay between the quadratic energy forms and a set of standardized orthonormal input stimuli may be explicitly characterized by a doubly stochastic matrix. This, in turn, leads to a number of interesting and useful inequalities between energies and eigenvalues of the system.