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 link between the two concepts is the convexity of quadratic forms associated with nonnegative 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.