Unitary extensions of isometries, generalized interpolation and band extensions

Abstract

The aim of this paper is to give a very brief account of some applications of the method of unitary extensions of isometries to interpolation and extension problems. I. Unitary extensions of isometries. A general method for solving several moment and interpolation problems can be summarized as follows: the data of the problem define an isometry, with range and domain in the same Hilbert space, in such a way that each unitary extension of that isometry gives a solution of the problem. In this review paper, the method and some of its applications are briefly described. We now fix the notation and then specify the content of the following sections. Unless otherwise specified, all spaces are assumed to be separable complex Hilbert spaces and all subspaces are closed; L(X,Y ) denotes the set of all bounded linear operators from a space X to a space Y ; L(X) is the same as L(X,X), and “ ∨ ” means “closed linear span”; P E ≡ PE denotes the orthogonal projection onto the subspace E of X and iE ≡ iE is the inclusion of E in X. L(X) denotes the space of X-valued measurable functions on the unit disk T with finite p-norm. L(X,Y ) denotes the space of L(X,Y )-valued measurable functions on T with finite p-norm. The isometry V acts in the Hilbert space H if its domain D and range R are (closed) subspaces of H. We say that (U,F ) belongs to U , the set of equivalence classes of minimal unitary extensions of V , if U ∈ L(F ) is a unitary extension of V to a space F that contains H, such that F = ∨ {UH : n ∈ Z}; we consider two minimal unitary extensions to be equivalent, and write (U,F ) ≈ (U ′, F ′) in U , if there exists a unitary operator X ∈ L(F, F ′) such that XU = U ′X and that its restriction to H equals the identity IH in H. An element (U,F ) of U with special properties is given by the minimal unitary dilation U ∈ L(F ) of the contraction V PD ∈ L(H). In Section II an isometry V is associated with a generalized interpolation problem in such a way that there is a bijection between U and the set of all the solutions of 1991 Mathematics Subject Classification: 47A57, 47A20.