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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 nite-dimensional system described by _ x = Ax +Bu, y = Cx +Du would be the s-dependent matrix S(s )= h A sI C B D i . In the general case, S(s) is an unbounded operator, and we show that it can be split into four blocks, as in the nite-dimensional case, but the splitting is not unique (the upper row consists of the uniquely determined blocksA sI andB, as in the nite-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 )w here 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 1 .W e also introduce the Lax-Phillips semigroup T 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 diers in several respects from the classical one, for example, by allowing an index !2 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 )i s not invertible, in terms of the spectrum of the generator of T (for various values of !). The system is dissipative if and only if T (with index zero) is a contraction semigroup.

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