CHAPTER 1
Overview
In this monograph, we will discuss the invariant subspaces of the backward
shift operator on the Hardy space. Here, for p G (0,oo), the Hardy space Hp is
the set of analytic functions / on the open unit disk D = {\z\ 1} for which the
quantity
Jo
27r J
is finite. The backward shift operator B on Hp is the continuous linear operator
defined for a function / = X ^ o
anzTl
£
Hp
by
Bf : =
/ ~ /(°)
= ai + a2Z + a 3 Z
2
+
.
m m
z
By "invariant subspace" for the operator B, we mean a closed linear manifold (i.e.,
a subspace) M of Hp for which BM. C A4. Although there are various aspects of
the backward shift on the Hardy spaces that are certainly worthy of attention, this
book will focus on only one of these topics:
the characterization of the backward shift invariant subspaces.
For p G (l,oo) the 5-invariant subspaces of Hp were described in a well known
paper of R. Douglas, H. S. Shapiro, and A. Shields [29]; while for p G (0,1], they
were described by A. B. Aleksandrov [3] in a remarkable paper which was never
translated into English and as a result, is not readily available in the West. The
main purpose of this book is to give a thorough treatment of all this material,
replete with the appropriate background material and full details of the proofs.
Although the backward shift operator is interesting to study in its own right,
it has important connections to the general study of bounded linear operators on
Hilbert spaces. Work of Rota [76], de Branges and Rovnyak [26], and Foia§ [36]
show that the restriction of a direct sum of backward shifts on H2 to an invariant
subspace is often the "model" for important classes of bounded linear operators. For
example, Rota showed that every strict contraction (a bounded operator T from a
Hilbert space to itself with operator norm ||T|| strictly less than one) is similar to
a direct sum of backward shift operators on
H2
restricted to an invariant subspace
(of the direct sum). One often uses the phrase T is similar to "part of a direct sum
of backward shifts". de Branges and Rovnyak, and Foia§ brought this model theory
to fruition and proved that if T satisfies the two conditions
||T|| 1 and
||Tnx||
- 0 a s n ^ o o Vx,
then T is unitarily equivalent to part of a direct sum of backward shifts. We refer
the reader to [65] [75] [98] for a more detailed treatment of model theory and its
connections to backward shifts.
1
http://dx.doi.org/10.1090/surv/079/01
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