Infinite networks of positive operators

Abstract

AbstractInfinite resistive networks have recently been analyzed by H. Flanders. The present work is an extension of Flanders results in three directions: It analyzes the more general case where the resistances in the network are positive bounded linear operators on a Hilbert space H and the currents and voltages are members of H. An infinite number of sources is now allowed. The network is permitted to have finite loops of doubly infinite paths consisting entirely of short circuits. Conditions are developed under which existence and uniqueness theorems for the set of branch currents can be stated.These conditions include the requirement that all branch resistances be either zero or invertible. The situation where some or all of the resistances have nontrivial null spaces is also considered. In those cases where the branch currents need not be unique for a given set of sources, the difference between two possible current distributions can be characterized as a current distribution that produces zero voltage drops in all branches. An application to time‐varying resistive networks containing switches is also given.