Modelling by L²-Bounded Analytic Functions

Abstract

Let E, F be two separable Hilbert spaces. We shall denote as in [5] by {E, F,θ(λ)} an analytic function defined in the open unit disk D of the complex plane with values bounded operators from E into F. We say that {E,F,θ(λ)} is inner if there exists, a.e. with respect to Lebesgue measure on the unit circle T, the strong limit

$$\theta \left( {{\text{e}}^{{\text{it}}} } \right) = \mathop {\lim }\limits_{x \to {\text{l}}} \theta \left( {{\text{re}}^{{\text{it}}} } \right)$$

and θ(e^it) is an isometry for almost all tε[0,2π].