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The concept of a Hilbert network, introduced in another paper, extends network theory to finite or infinite networks whose elements are described by operators on a Hilbert space. The present work investigates a variety of qualitative properties possessed by such networks. In particular, some operators associated with the entire network, such as the driving-point impedances as well as the admittance operator which relates the branch-voltage vector to the branch-current vector, are shown to be—under suitable conditions—either positive, monotonic, or convex. Also, generalized versions of Jeans’ least power theorem and the Shannon–Hagelbarger theorem are proved. Finally, a bound on the power dissipation is determined.

In this paper there is constructed a simple model of a general nonlinear network. The network considered consists of at most countably many lumped elements described by (not necessarily linear) operators from a Hilbert space into itself. Several theorems are proved on the existence and uniqueness of the solution of the network. Also, it is shown that, under certain conditions, the solution of an infinite network is a limit of solutions of finite subnetworks.

This paper contains an abstract treatment of functional equations which involve non-linear causal operators. We consider equations on linear spaces that are certain extensions of Banach or Hilbert spaces. The causality is defined in the traditional way, but by a “past” we mean a set from a given family of subsets of a generalized “time domain.”An operator A is said to have a resolvent operator Q, if for each u from a certain space, the equation $x = A(x,u)$ possesses a unique solution x ; then $x = Qu$. We prove a theorem giving conditions for the existence of Q, and for continuity and boundedness of Q in a certain sense.Further results are obtained for the case $A(x,u) = Bx + u$, i.e., for an equation of Hammerstein’s type. In particular, it is shown that some monotonicity properties of operators involved guarantee the existence and continuity of the resolvent Q. The theory is illustrated by several concrete examples.

This paper presents a theorem stating conditions under which an n-port obtained by series and parallel connections of elementary n-ports is passive, i.e., incapable of. producing energy under any excitation. An electrical n-port built up from purely inductive, resistive and capacitive time-varying n-ports by combining them in series and parallel is a type of the n-port considered.

The paper deals with a qualitative property of linear continuous operators defined on distributions, that is, with a property appearing as a generalization of the traditional causality. The traditional causality of an operator A means that ${\operatorname{supp}}Ax \subset S$ whenever ${\operatorname{supp}}x \subset S$, where A is any interval $[T,\infty )$. Here, causality is defined in the same way, but S is any member from a certain family of subsets of $R^m $, called a scale.A theorem is proved which gives necessary and sufficient conditions for an operator to be causal with respect to a given scale.As an application there is considered a slightly generalized convolution-type operator; it is shown that this operator is causal with respect to a scale which consists of all translations of a fixed cone.