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