Almost Definite Operators and Electro-mechanical Systems

Abstract

In recent years the concept of parallel addition and other new operations have been introduced. These operations are derived from network interconnections involving Kirchhoff’s and Ohm’s laws. From electrical considerations the domains of these operations, for the most part, have been restricted to positive semi-definite operators on a finite dimensional inner product space. In this paper we consider general electro-mechanical systems of the form: given a complex linear transformation $A:\mathcal{K} \to \mathcal{W}$ and a complex linear operator Z on $\mathcal{U}$, we seek a linear operator $\Psi ( Z )$ on the range of A, such that for any $b \in {\operatorname{rang}} ( A )$, there is a solution x, v to $Ax = b$, $Zx - A^ * v = 0$ with $\Psi ( Z )b = v$. We term Zalmost definite if $( {Zx,x} ) = 0$ only when $Zx = 0$. If Z satisfies this condition, then $\Psi ( Z )$, termed the transfer resistance, exists and is unique. Various properties are developed. In particular, An operator Z is almost definite if and only if $\Psi ( Z )$ exists for all possible A. If Z is almost definite, then there is a complex constant $\alpha $ with $| \alpha | = 1$, such that Re $( {\alpha Zx,x} ) \geqq 0$, for all $x \in \mathcal{U}$. This model extends the domain of validity of the new operations and unites the theory.