Transfer functions of regular linear systems. Part II: The system operator and the Lax-Phillips semigroup

This paper is a sequel to a paper by the second author on regular linear systems (1994), referred to here as "Part I". We introduce the system operator of a well-posed linear system, which for a finite-dimensional system described by ẋ = Ax + Bu, y = Cx + Du would be the s-dependent matrix S∑(s) = [C^A-sID^B]. In the general case, S∑(s) is an unbounded operator, and we show that it can be split into four blocks, as in the finite-dimensional case, but the splitting is not unique (the upper row consists of the uniquely determined blocks A-sI, and B, as in the finite-dimensional case, but the lower row is more problematic). For weakly regular systems (which are introduced and studied here), there exists a special splitting of S∑(s) where the right lower block is the feedthrough operator of the system. Using S∑(0), we give representation theorems which generalize those from Part I to well-posed linear systems and also to the situation when the "initial time" is -∞. We also introduce the Lax-Phillips semigroup T-fraktur sign induced by a well-posed linear system, which is in fact an alternative representation of a system, used in scattering theory. Our concept of a Lax-Phillips semigroup differs in several respects from the classical one, for example, by allowing an index ω ε ℝ which determines an exponential weight in the input and output spaces. This index allows us to characterize the spectrum of A and also the points where S∑(s) is not invertible, in terms of the spectrum of the generator of T-fraktur sign (for various values of ω). The system ∑ is dissipative if and only if T-fraktur sign (with index zero) is a contraction semigroup.