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Chapter 1

Introduction

Contents

1.1 Basic Definitions and Results . . . . . . . . . . . . . . 1

1.2 Thesis overview . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.1 Invariance in BMOA . . . . . . . . . . . . . . . . . . 7

1.2.2 Lax-Halmos type theorems in Hpspaces . . . . . . . . 8

1.2.3 Invariance in Certain Sub-Hilbert spaces of H2. . . . 10

1.1 Basic Definitions and Results

Let T stand for the unit circle in the complex plane, and D stand for the open

unit disk in the complex plane. For 0 < p < ∞, Hp stands for the class of

analytic functions f(reit) on D satisfying the growth condition:

�f�p := sup0<r<1

1

2π

�� 2π

0

|f(reit)|pdt�1

p< ∞. (1.1)

By H∞ we mean the class of bounded analytic functions on D, that is, all

analytic functions f(eit) on D satisfying the condition:

�f�∞ := esssup�|f(reit)| : reit ∈ D

�< ∞. (1.2)

2 Chapter 1. Introduction

The spaces Hp are called Hardy spaces, and for 0 < p ≤ ∞ they are

Banach spaces under the norm given by equations (1.1) and (1.2). Also, H∞

is a Banach algebra under the pointwise product of functions, and H2 is a

Hilbert space under the inner product

< f, g >2:= limr→1

1

2π

� 2π

0

f(reit)g(reit)dt. (1.3)

Let Lp denote the familiar Lebesgue spaces on the unit circle. It is well

known that a function on the class Hp for any p has non-tangential boundary

values almost everywhere on the unit circle and that the boundary function

lies in the class Lp. Further, for 1 ≤ p ≤ ∞, the negative Fourier coefficients

of the boundary function will vanish, that is, if f is the boundary function

then � 2π

0

f(reit)e−intdt = 0 (1.4)

for n = −1,−2, . . .. Conversely, if we start with any function in Lp for p ≥ 1

then such a function can be extended to the open unit disk by means of the

Poisson kernel and this extended function will satisfy equation (1.1) if p < ∞,

and equation (1.2) if p = ∞. This correspondence actually establishes an

isometric isomorphism between the Hp spaces as defined by equations (1.1)

and (1.2) and the class of functions in Lp satisfying condition (1.4). Thus

we make no distinction between the class of functions in Hp and the class of

functions in Lp whose negative Fourier coefficients vanish, as denoted by (1.4).

So we shall denote both these classes by Hp.

The space H2 can also be thought of as a space of formal power series

whose coefficients form a square summable sequence. H∞ can be realized as

all those f(z) in H2 such that f(z)g(z) is in H2 for all g(z) in H2, and �f�∞

1.1. Basic Definitions and Results 3

is also given by sup{�fg�2 : �g�2 = 1}.

It is well known that every f(z) ∈ Hp can be written as

f(z) = B(z)S(z)O(z) (1.5)

where

B(z) = zm∞�

n=1

|an|(an − z)

an(1− anz)

S(z) = exp

�−�

eit + z

eit − zdµ(t)

�

Here ans are the zeros of f in the disk repeated according to multiplicity

and µ is a positive Borel measure on T singular with respect to the Lebesgue

measure. Further

O(z) = α exp

�1

2π

�

T

eit + z

eit − zlog |f(eit)|dt

�.

The functions B(z) and S(z) are functions of unit modulus almost everywhere

on the unit circle.

Any function I(z) in Hp which is of unit modulus almost everywhere with

respect to the Lebesgue measure on T is called an inner function. It can be

proved that a function I(z) is inner if and only if {I(z)zn : n = 0, 1, 2 . . .} is

an orthonormal set in H2. A function of the form of O(z) is called an outer

function. The inner-outer factorization stated in equation (1.5) is unique.

There is a very important Banach space of functions contained inside every

Hardy space. This space is the class of analytic functions of bounded mean

oscillation and is denoted by BMOA. The elements of BMOA are all f ∈ H1


such that

�f�∗ := sup1

|I|

�

I

��f − 1

|I|

�

I

f

�� < ∞ (1.6)

where the supremum is taken over all subarcs I of the unit circle and |I|

denotes the normalized Lebesgue measure of I. BMOA is a Banach space

under the norm

�f� := �f�∗ + |f(0)|. (1.7)

Now we state the famous theorem of Fefferman which establishes the duality

between BMOA and the Hardy space H1.

Theorem 1 (Fefferman’s theorem) BMOA is the dual of H1 and the ac-

tion of any BMOA function f treated as a functional on H1 is given by

f(p) =1

2π

�

Tf(eit)p(eit)dt

where p is any polynomial in H1.

Let H be a Hilbert space and T be a bounded linear operator on H. A

proper non trivial closed subspace M of H is called an invariant subspace of

T if T (M) ⊂ M . Invariant subspaces have been of great interest for several

reasons (see [14], [25]). The invariant subspace problem asks if any operator

on an infinite dimensional separable Hilbert spaces has an invariant subspace.

One of the first invariant subspace theorems of fundamental importance was

the theorem of Beurling [2]. This theorem characterized the invariant sub-

spaces of the operator S of multiplication by the coordinate function z on the

Hardy space H2. Beurling showed that the invariant subspaces of S are of

the form ϕH2, where ϕ is an inner function. One of the first generalizations

of this was due to Lax [19]. In a unitarily equivalent form Lax’s theorem

1.1. Basic Definitions and Results 5

characterizes the invariant subspaces of the operator of multiplication by the

coordinate function z on the Hardy space H2(Ck) of Ck valued functions on

the disk D.

Another generalization of the Beurling’s theorem is the famous result of

de Branges [33]. This result also generalizes the Lax-Halmos theorem. In

the scalar context, de Branges theorem characterizes the class of all Hilbert

spaces that are contractively contained in the Hardy space H2 and on which

the multiplication by the coordinate function acts as an isometry. A Hilbert

space H is said to be contactively contained in a Hilbert space K if H is a

vector subspace of K and �x�K ≤ �x�H for all x ∈ H.

Singh and Singh [37] have proved a general version of the de Branges

theorem by dropping the condition of contractive containment. Singh and

Agrawal [36] extend this general version of de Branges theorem to Hp for all

1 ≤ p ≤ ∞. On H2, a general version of the de branges result was proved

by Singh and Thukral [39] by considering multiplication by finite Blaschke

factors instead of multiplication by the coordinate function z. Some of the

ideas of Singh and Thukral [39] have been used by Lance and Stessin [18] to

characterize all subspaces of Hp (for all p ≥ 1) that are invariant under a

finite Blaschke factor.

Two variable versions of de Branges theorem have also been obtained.

First, Singh [35] obtained a characterization on H2(T2), and then Redett [29]

extended Singh’s result to Hp(T2) for p ≥ 1. Both authors assume that

multiplication by coordinate functions are doubly commuting isometries on

the Hilbert space in question.

Recently, Paulsen and Singh [24] have proved a Beurling type result that

characterizes the common invariant subspaces of z2 and z3 in Hp (p ≥ 1).


Raghupathi et. al. [7] provide an alternative proof of Paulsen and Singh’s

result for the Hardy space H2. This invariant subspace characterization is fur-

ther used in [7] to obtain necessary and sufficient condition for a Nivanlinna-

Pick type interpolation result to hold on the unit disk with the additional

assumption that the derivative of all analytic interpolating functions vanishes

at the origin. In the present thesis we will present a recent work of Sahni and

Singh [31] where the invariant subspace results of [7] and [24] have been gen-

eralized by replacing multiplication by z by multiplication by a finite Blaschke

factor B(z). This result is valid for Hp for all 0 < p ≤ ∞. In [31], the au-

thors also extend the invariant subspace characterization of Lance and Stessin

[18] to the case of Hp where 0 < p < 1 and provide an alternative simpler

proof (which does not depend on the general inner-outer factorization of Hp

functions) for the cases 1 ≤ p ≤ ∞.

Invariant subspaces have been an area of vigorous research for the past

decades because of their use in solving Nevanlinna-Pick interpolation type

problems for certain algebras. In particular the algebras H∞ and H∞1 have

grabbed most attention. For further details we refer the reader to [6], [7], [16],

[21], [24] and [27]. The starting point in the sequence of papers cited above is

an invariant subspace result first proved in a special case in [24] and then in

more general forms in [7] and [26].

1.2 Thesis overview

The research carried out in the thesis is divided into three chapters titled

Invariance in BMOA, Lax-Halmos type theorems in Hp spaces, and Invari-

ance in Certain Sub-Hilbert spaces in H2. A summary of each of the above

1.2. Thesis overview 7

mentioned chapters is as follows:

1.2.1 Invariance in BMOA

Let S : BMOA −→ BMOA such that S(f(z)) = zf(z). The S invariant

subspaces of BMOA were first obtained by Singh and Singh [38]:

Theorem 1.2.1 (Theorem C, [38]) Let M be a weak star closed subspace

of BMOA invariant under S. Then corresponding to this M there exists a

unique inner function ϕ and a unique subspace N of BMOA such that N is

weak star dense in BMOA and M = ϕN . Further N = BMOA if and only

if ϕ is a finite Blaschke product.

In the proof of Theorem 1.2.1, the authors first characterize the backward

shift invariant subspaces of H1, that is, those closed subspaces of H1 which are

invariant under the operator S∗ : H1 −→ H1 defined by S∗(f) = z(f − f(0)):

Theorem 1.2.2 (Theorem 3.1, [38]) Let M be a closed subspace of H1 in-

variant under S∗. Then there exists a unique inner function φ such that

M = φH10 ∩H1.

They, then use the Fefferman’s duality theorem to derive the S invariant

subspaces of BMOA.

In this chapter we present a recent work of Sahni and Singh [32], where

the above steps are reversed by first obtaining the proof of Theorem 1.2.1

directly by fairly elementary means; and then this characterization is used

to re-discover Theorem 1.2.2. Lastly, using Theorem 1.2.2, we present the

characterization of S invariant subspaces of BMOA in an alternative form

(also discovered by Brown and Sadek [3]):


Theorem 1.2.3 (Theorem 4.3, [32]) Let M be a weak star closed S-

invariant subspace of BMOA. Then sycorresponding to this M there exists a

unique inner function ϕ such that M = ϕ · BMOA ∩BMOA.

Using similar ideas, we derive a new characterization of the common in-

variant subspaces of S2 and S3 on BMOA. The main result is as follows:

Theorem 1.2.4 (Theorem 3.1, [32]) Let M be a weak star closed subspace

of BMOA which is invariant under S2 and S3 but not invariant under

S. Then there exists an inner function I, and constants α, β such that

M = IBMOAα,β ∩ BMOA. (BMOAα,β is the weak star closure in BMOA

of span{αz + β, z2BMOA})

1.2.2 Lax-Halmos type theorems in Hp spaces

This chapter is devoted to the study of closed subspaces (weak star closed if

p = ∞) of Hp which are invariant under multiplication by a finite Blaschke

factor

B(z) =n�

j=1

z − αj

1− αjz;α1 = 0.

We shall call such subspaces as B-invariant. For the cases when 1 ≤ p ≤ ∞,

Lance and Stessin [18] have characterized the B-invariant subspaces of Hp.

Their proof relies on the general inner-outer factorization of Hp functions

which they establish first in [18]. Here we present a new proof by Sahni and

Singh [31] that is ‘factorization free’. The method adopted has led to a much

simpler proof and has enabled us to extend the result to the cases 0 < p < 1

as well:

Theorem 1.2.1 (Theorem 4, [31]) Let M be a closed subspace of Hp, 0 <

p ≤ ∞, p �= 2, such that M is invariant under H∞ (B). Then there exist B−


inner functions J1, . . . , Jr, r ≤ n, such that

M = J1Hp (B)⊕ · · ·⊕ JrH

p (B) .

When 0 < p < 1, then, as the proof will show, the right hand side is to be read

as being dense in M i.e. its closure in the Hp metric is all of M .

Next, we use similar ideas to characterize the common invariant subspaces

of B2 and B3. Our result runs as follows:

Theorem 1.2.2 (Theorem 3, [31]) Let M be a closed subspace of Hp, 0 <

p ≤ ∞, such that M is invariant under H∞1 (B) but not invariant under

H∞ (B). Then there exist B− inner functions J1, . . . , Jr (r ≤ n) such that

M =

�k�

j=1

⊕�ϕj��

⊕r�

l=1

⊕B2JlHp (B)

where k ≤ 2r− 1, and ϕj = (α1j +α2jB)J1 + (α3j +α4jB)J2 + ...+ (α2r−1,j +

α2r,jB)Jr, for all j = 1, 2, ..., k.

Remark. The proof shall also show that the matrix A = (αij)2r×ksatisfies

A∗A = I, and αst �= 0 for some (s, t) ∈ {1, 3, . . . , 2r− 1}× {1, 2, . . . , k}. Also,

when 0 < p < 1, the right hand side should be interpreted as being the closure

of the sum in the Hp metric.

Theorem 3.3.1 clearly generalizes the following invariant subspace theorem

obtained in [7] and [24]:

Corollary 1.2.3 (Paulsen & Singh[24]; Raghupathi[7]) Let M be a

closed subspace space of Hp, 0 < p ≤ ∞, and which is invariant under multi-

plication by z2 and z3 but not invariant under z. Then there exist scalars α, β


with α �= 0 and |α|2 + |β|2 = 1, and an inner function J , such that

M = �J(α + βz)� ⊕ z2JH2.

1.2.3 Invariance in Certain Sub-Hilbert spaces of H2

Let H stand for a Hilbert space that is a vector subspace of the Hardy space

H2 and satisfying the following axioms:

(A1) For any four functions f1, f2, f3, f4 ∈ H, satisfying �f1, g1�H = �f2, g2�H ,

we must have �f1, g1�H2 = �f2, g2�H2 .

(A2) If ϕ is any inner function; then ϕf ∈ H and

�ϕf,ϕg�H= �f, g�

Hfor all f, g ∈ H.

Recently, Yousefi and Hesamedini [41] have characterized closed subspaces of

H that are invariant under multiplication by z. Their main result runs as

follows:

Theorem 1.2.1 (Theorem 3, [41]) Let H be a Hilbert space contained in

H2 satisfying axioms A1 and A2. Let M be a closed subspace of H that is

invariant under the operator S. Further if the set of multipliers of H coincide

with H∞, then there exists a unique inner function ϕ such that M = ϕH.

The characterization of H, however, is left an open question: Is every

Hilbert space H that is algebraically contained in H2 and satisfying axioms

(A1) and (A2) of the form ϕH2 for some ϕ ∈ H∞? Also, does there exist a

constant k such that �f, g�H= k �f, g�

H2 for every f, g ∈ H?

In this chapter, a recent work of Sahni and Singh[30] is presented, wherein

the above open problem is settled in the affirmative and that too under much

weaker assumptions. The result runs as follows:


Theorem 1.2.2 (Theorem 3.1, [30]) Let H be a Hilbert space that is alge-

braically contained in H2 and satisfies the following axioms:

(i) TB (H) ⊂ H and �Bf,Bg�H= �f, g�

Hfor all f, g ∈ H;

(ii) If f1, f2, g1, g2 ∈ H satisfy �f1, g1�H = �f2, g2�H , then we have

�f1, g1�H2 = �f2, g2�H2.

Then there exist B-inner functions b1, b2, . . ., br (r ≤ n) such that

H = b1H2(B)⊕ · · ·⊕ brH

2(B).

In this result B stands for a finite Blaschke factor of order n, and the operator

TB stands for the multiplication by B on H2. Observe that assumption (i)

is weaker than the axiom (A2). For B(z) = z we have the following special

case:

Corollary 1.2.3 Let H be a Hilbert space that is algebraically contained in

H2 and satisfies the following axioms:

(i) Tz (H) ⊂ H and �zf, zg�H= �f, g�

Hfor all f, g ∈ H;

(ii) If f1, f2, g1, g2 ∈ H satisfy �f1, g1�H = �f2, g2�H , then we have

�f1, g1�H2 = �f2, g2�H2.

Then there exists a unique inner function b such that H = bH2 and there is a

constant k such that �bf�H = k�f�H2 for all f ∈ H2.

Theorem 4.1.1 is now re-discovered without the assumption that the mul-

tipliers of H coincide with H∞.


Theorem 1.2.4 (Theorem 3.3, [30]) Let H be a Hilbert space satisfying

the conditions of Corollary 1.2.3. Let M be a closed subspace of H which

is invariant under Tz. Then there exists a unique inner function ϕ such that

M = ϕH.

The proof of Theorem 1.2.4 uses Corollary 1.2.3 and is simpler than the

one obtained in [41].

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