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 of
H2
became to be known as model spaces
via the Nagy-Foia¸ s theory when the compression of the shift S to 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 (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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