Generalized couplings
Vlastimil Pták · Czechoslovak Mathematical Journal · 1995
The notion of generalized couplings was introduced and investigated, not long ago, in a joint paper of P. Vrbova and the present author [4].The idea of combining two spaces, each with an operator given on it, appears first-in the particular case of semiunitary operators-in a paper of Adamyan and Arov.The ideas of Adamyan and Arov [1] were further developed by a number of authors; in particular, the connections of this notion with dilations of mappings of positive type form the subject of related investigations of Arocena and Cotlar [2].In its full generality, as considered in [4], the problem may be formulated as follows.Given two bounded linear operators A\ and A 2 acting on the Hilbert spaces J?[ and Jff 2 and a contraction X: Jf?\ -• » J$? 2 , what are the conditions for the existence of a Hilbert space Jf and an operator U E B(J(f) such that (1) Jff contains Jtf[ and J^2 l (2) J%{ is U invariant and U\j4f{ = A\, (3) Jff 2 is U* invariant and U*\j¥ 2 = A 2 , (4) P(Jf 2 )\j{?\ =X.By P(Jf) we denote the orthogonal projection onto Jf.An operator satisfying the four properties listed above is called a coupling of A\ and A 2 .It is not difficult [4] to show that, for the existence of U, the following intertwining relation is necessary: the existence of U implies the relation XA\ = A\X.This necessary condition for the existence of couplings is also sufficient; it turns out that contractive U exist if A\ and A 2 are both contractions.In [4] a parametrization of all contractive couplings was given together with a characterization of the triples A\, A 2 , X for which an isometric coupling exists.The approach used in [4], though straightforward enough, does not permit to describe the parametrization in terms of the original spaces and operators in a simple manner; in particular, the verification of the existence conditions for isometric solutions is not simple.In view of the fact that many questions in dilation theory may be reformulated in terms of couplings of suitable spaces and operators