Preface

The main theme of this proceedings volume is the invariant subspaces of the

shift operator S, or its adjoint S∗, on certain reproducing kernel Hilbert spaces of

analytic functions on the open unit disk. Such spaces include the Hardy spaces H2,

the Dirichlet space D, the de Branges-Rovnyak spaces H(b), and the model spaces

KΘ.

Model spaces have many fascinating aspects. For one, they represent, via Beurl-

ing’s theorem, which characterizes the invariant subspaces of the shift operator

Sf = zf on

H2,

the complete set of invariant subspaces of the backward shift

operator

S∗f

= (f − f(0))/z. Using the theory of pseudo continuations developed

by H. S. Shapiro, the description of these backward shift invariant subspaces was

described in a seminal paper of Douglas, Shapiro, and Shields. The concept of pseu-

docontinuation continues to have an uncanny way of appearing in many unexpected

areas of analysis.

These backward shift subspaces KΘ of

H2

became to be known as model spaces

via the Nagy-Foia¸ s theory when the compression SΘ of the shift S to KΘ was

shown to represent a wide class of contraction operators on Hilbert space. Though

these spaces make connections to operator theory and some areas of mathematical

physics, they are a fascinating Hilbert space of analytic functions in the own right.

They have interesting reproducing kernels and boundary behavior. The rank-one

unitary perturbations of SΘ (the Clark unitary operators) are an interesting class of

operators with an even more interesting and useful set of spectral measures (Clark

measures).

Parallel to the theory of Sz.-Nagy and Foias, de Branges and Rovnyak developed

another model based on H(b) spaces, where b is an analytic function in the closed

unit ball of H∞ (bounded analytic functions on the open unit disk). When b is

inner, H(b) coincides with the model space Kb. In the general case, H(b) spaces

are not a closed subspace of

H2,

but they are equipped with a norm to become

a Hilbert space contractively embedded in

H2.

Their structure is fascinating and

depends on whether or not log(1 − |b|) is integrable on the unit circle, equivalently,

non-extreme or extreme points in the closed unit ball of

H∞.

Since the foundation

developed by de Branges and Rovnyak, H(b) spaces continue to be a precious

tool in various questions in analysis such as function theory (resolution of the

Bieberbach conjecture by de Branges, rigid functions, Schwarz–Pick inequalities),

operator theory (invariant subspace problem, composition operators, kernel of the

Toeplitz operators), systems and control theory.

Related to the Hardy space, model spaces, and de Branges-Rovnyak spaces,

is the Dirichlet space. This space connects to many areas and tools of analysis.

For example, the definition of the Dirichlet space (analytic functions on the disk

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