Preface

Summary

In 1915, Godfrey Harold Hardy, in a famous paper published in the Proceedings of the London Mathematical Society, first put forward the “theory of Hardy spaces” H^p. Having a Hilbert space structure, H² also benefits from the rich theory of Hilbert spaces and their operators. The mutual interaction of analytic function theory, on the one hand, and operator theory, on the other, has created one of the most beautiful branches of mathematical analysis. The Hardy–Hilbert space H² is the glorious king of this seemingly small, but profoundly deep, territory.

In 1948, in the context of dynamics of Hilbert space operators, A. Beurling classified the closed invariant subspaces of the forward shift operator on l². The genuine idea of Beurling was to exploit the forward shift operator S on H². To that end, he used some analytical tools to show that the closed subspaces of H² that are invariant under S are precisely of the form ΘH², where Θ is an inner function. Therefore, the orthogonal complement of the Beurling subspace ΘH², the so-called model subspaces KΘ, are the closed invariant subspaces of H² that are invariant under the backward shift operator S^∗. The model subspaces have rich algebraic and analytic structures with applications in other branches of mathematics and science, for example, control engineering and optics.

The word “model” that was used above to describe KΘ refers to their application in recognizing the Hilbert space contractions. The main idea is to identify (via a unitary operator) a contraction as the adjoint of multiplication by z on a certain space of analytic functions on the unit disk. As Beurling's theorem says, if we restrict ourselves to closed subspaces of H² that are invariant under S^∗, we just obtain K_Θ spaces, where Θ runs through the family of inner functions. This point of view was exploited by B. Sz.-Nagy and C. Foiaş to construct a model for Hilbert space contractions. Another way is to consider submanifolds (not necessarily closed) of H² that are invariant under S^∗. Above half a century ago, such a modeling theory was developed by L. de Branges and J. Rovnyak. In this context, they introduced the so-called H(b) spaces.