On the Line and in the Halfplane

Abstract

All the previous Lectures were devoted to the study of the semigroup Lⁿ_n≖0 generated by the shift operator L in the space L²(T). Also the continuous analogue of this theory, connected with the space L²(ℝ), is useful, in particular in scattering theory and in the study of stationary stochastic processes. The semi-group of shifts in L²(ℝ) is interesting also from the point of view of function theory and we shall reduce the study of it to results already established for the operator L, limiting ourselves however only to the proof of Lax’s fundamental theorem on the description of invariant subspaces and to a transplantation of a small number of the results of Lecture I-VIII into the language of the ℝ-theory. Concerning the details of the theory of semigroups of contractions obtained by projecting the shifts onto coinvariant subspaces, the reader is referred to the books and papers listed in the “Concluding Remarks”.