The abstract inverse scattering problem and the instability of completeness of orthogonal systems (Q1114138)

This is a selfcontent and clearly written paper on the abstract inverse scattering problem. The main theorem proved by the author states that every contractive operator-valued function S(t), \(t\in T\) (unit circle) on the Hilbert space E is the scattering operator for a pair \((U,\overset\circ U)\) of unitary operators on the Hilbert space \(L^ 2(E)\) of E-valued square-integrable functions on T with \((\overset\circ U f)(t) = tf(t)\), \(t\in T\). The tools used by the author include his generalization of Weyl's criterion of essential not left invertibility of an operator, the general theory of orthogonal systems and the scattering theory.