Causality in Hilbert Space

Abstract

A new formalism, termed the resolution space, is presented within which the theory of causal systems may be unified and extended. The resulting formalism, which is defined as a Hilbert space together with a resolution of the identity, readily includes the commonly encountered function and sequence space causality concepts yet is sufficiently straightforward to allow the various aspects of network and system theory which are dependent on the time parameter to be studied in an operator theoretic context without the detailed structure of a function space. Specific results include additive and multiplicative decomposition theorems for causal operators which naturally extend the "realizable part" and "spectral" decompositions of classical system theory and an integral representation theorem for linear operators on a resolution space. The general theory is illustrated with a number of examples concerning passive "networks", those including an operator theoretic approach to the passive synthesis problem over an arbitrary resolution space.In the various applications a number of different, but essentially equivalent, formulations of the causality concept have been given. These, of course, include the usual function and sequence space approaches (wherein operator [14], [34] and Hilbert space valued [3], [6], [12], [35], [37] functions are employed) as well as a number of invariant subspace formulations 7], 15], [21], [25]. Possibly *

Causality Structure of Engineering Systems

“…The concepts of causal, anticausai, and memoryless systems are in formal agreement with those proposed by Porter [ 421], [ 43 ] and Saeks [ 48 ]. It will be apparent later that this agreement extends also to the concepts of strictly causal and strictly anticausal systems.…”

“…Recently this new type of approach has been supported by a number of other authors ll1 [42], [43], [48], 160]. Here the works of Porter, and of Saeks are the most closely related to the present study and will be a subject of attention later in this development.…”

“…The development of this chapter can perhaps be best introduced by a brief discussion of some relevant work of Saeks [ 48 ].…”

“…Apparently this situation has been justified by the feeling that noncausal behavior can be treated as causal behavior in reverse time. This suits quite well studies confined to physical systems and, to a cortain extent, linear systems [ 48 ]. However, such an approach is not applicable to the type of problems considered here.…”

“…A more general approach has recently been adopted by Porter [41], [ 42] and Saeks [ 48], [49 1. These references treat causal and noncausal systems on an equal footing and provide much of the ideas and inspiration for the present development.…”