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1193 campus-p.14

 

THE PRODUCT OF A
COMPOSITION OPERATOR
WITH THE
ADJ OINT OF A COMPOSITION OPERATOR

By

John Howard Clifi’ord

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

Department of Mathematics

1998

ABSTRACT

THE PRODUCT OF A
COMPOSITION OPERATOR
WITH THE
ADJ OINT OF A COMPOSITION OPERATOR

By

John Howard Clzfio'rd

We obtain an upper estimate for the essential norm of 03,0“, on the Hardy space
H 2 as the upper bound of a quantity involving the product of the inducing maps’
Nevanlinna counting functions. In the special case of univalent inducing maps we
prove a complete function theoretic characterization of compactness in terms of the
angular derivatives of the inducing maps.

We obtain necessary and sufficient conditions, under varied hypothesis on the
inducing maps, for the operator CPU}; to be compact on the Hardy space H 2. In the
special case where one inducing map is boundedly valent we calculate a lower estimate
for the essential norm as the upper bound of a quantity involving the product of the

inducing maps’ Nevanlinna counting functions.

To Joan

iii

ACKNOWLEDGMENTS

I would like to express my sincere gratitude to my advisor, Professor Joel Shapiro,
for his excellent guidance, encouragement, and advice in the preparation of this doc-
ument. I am extremely grateful for the many outstanding courses he taught that I
was fortunate enough to attend.

I would also like to express my sincere appreciation to Dechao Zheng for his
support, insights, and for the many enjoyable hours we spent talking about the ideas
in this document.

iv

TABLE OF CONTENTS

INTRODUCTION

1 Linear Fractional Maps
1.1 Comparison principle. ..........................

2 Preliminaries
2.1 An equivalent inner product on H 2 ....................
2.2 Reproducing kernels and essential norm. ................
2.3 Nevanlinna counting function. ......................
2.4 Change of variables formula ........................
2.5 Littlewood’s Inequality ...........................
2.6 Angular derivative. ............................

3 The Angular Derivative and
the Essential Norm

4 The Operator 0,209,, on H 2
4.1 Main results for Cng ..........................
4.2 Upper estimate on the essential norm of 0,7,6}? ............
4.3 Necessary condition for 0,20,, to be compact. .............

5 The Operator 047;; on H 2
5.1 Main results for Cng. ..........................
5.2 Lower estimate for the essential norm of GPO; ............
5.3 Sufficient condition for 6'ng to be compact ...............

BIBLIOGRAPHY

11

13
13
14
15
16
17
18

21

26
26
28
34

38
38
40
43

53

Introduction

Let U denote the open unit disc of the complex plane and let cp and d be holomorphic

self maps of the disk. The equation

quf=f°90

defines a composition operator C,p on the space of holomorphic functions; so is called
the inducing map or symbol of CW It is a consequence of Littlewood’s subordination
principle [8] that 0,, is bounded on H2. This paper studies the compactness of the
operators formed by multiplying a composition operator C“, with the adjoint 0:], of
another composition operator to form either GPO; or 0,7,6}. Our goal is to give a
function theoretic characterization of the essential norms of C¢CJ, and 03,099 in terms
of the geometric properties of the inducing maps (p and 1/2. This line of investigation
has already been carried out for composition operators acting on the classical weighted
Hardy and Bergman spaces. Let IITIIe denote the essential norm of the Operator T
on H2 (i.e the distance in the operator norm from T to the compact operators).

Shapiro [16] gave the following expression for the essential norm of C,p on H 2,

 

N¢(w)
logI—l—I’ (1)

“Cw”: = lim 811p
le—rl"

thus providing a complete function theoretic characterization of compact composition

operators in terms of the inducing map’s Nevanlinna counting function Nw.

We make progress toward answering in the affirmative the following two conjec-

tures:

Conjecture 1 Suppose go and w are holomorphic self maps of the disc. Then

 

HHC‘HCcpHH: : limsup IY¢(‘p(f))iV¢(z/)l(z)) (2)
|z|—)l"' 08m Ogm

Conjecture 2 Suppose go and 1,0 are holomorphic self maps of the disc. Then

 

N w N w
“not“: = 1imSU_p t‘ l 1‘“. l. (3)
|w|->l (log H7!)
The study of compact composition operators on H 2 first appeared in
H. J. Schwartz’s [14] thesis in the late sixties. He proved a necessary condition for a

composition operator to be compact;
If C,p is compact then [90*] < 1 a.e. on the unit circle.

In other words, C“, is not compact whenever the set {Icp*| = 1} has positive measure.
We let go“ denote the nontangential limit (when it exists) of 90. (By Fatou’s theorem
this limit exists at almost every point of 6U). Schwartz also showed this necessary

condition is not sufficient by showing the composition operator induced by

 

is not compact, even though cp maps only a single point of the unit circle onto the
unit circle, <p(1) = 1. This work was carried on in [15] by Shapiro and Taylor who

extended Schwartz’s theorem as follows:

Cg, is not compact whenever (,0 has an angular derivative at some point

of the unit circle.

In [14], Schwartz also proved the following sufficient condition, which we will refer

to as the Ll-condition, for a composition operator to be compact.
If (1 -— |c,o"‘(e“)|)"l E L1(8U) then C"p is compact.

Shapiro and Taylor [15] refined this result by showing that the Ll-condition charac-
terizes the Hilbert-Schmidt composition operators on H 2. Moreover, they applied the
Ll-condition to show that if the image of go is contained inside a polygon then Co is
Hilbert-Schmidt.

In [9] MacCluer and Shapiro studied the extent to which the angular derivative
characterizes the compactness of a composition operator. They obtained a sharper

result than we state (i.e. [9], Theorem 3.10) but for our purposes we highlight;

Suppose cp is finitely valent. Then C“, is compact if and only if (,0 has no

angular derivative.

In [9] it is shown that on the weighted Bergman spaces the angular derivative does
tell the whole story, i.e. for any self map (,0 of the unit disc, CV, is compact if and only
if cp has no angular derivative.

In 1987, Shapiro [16] solved the compactness problem by deriving the essential
norm formula,
N <p(w)

log|—1—l'

 

“Cw“: = lim sup
[w|—)l"

Hence,

Cw is compact if and only if lim M =
[ml—)1" 10g m

This paper is organized as follows. In the next chapter we consider the operators
(3'ng and 0,00,)“, induced by linear fractional self maps of the disc. As motivation

for more general results we completely characterize the compactness of these linear

fractionaly induced operators in terms of the inducing maps. Chapter 3 consists of a
sketch of background material. In Chapter 4 we develop the function theory needed
to connect the compactness of the operators with the angular derivatives of <p and
2/2. The next two chapters treat the operators (7,130,p and CwCJ, respectively. In the
first section of each these chapters we outline the main results about the operator in
question and develop the connection between the compactness of that operator and

the angular derivative of (p and it).

CHAPTER 1

Linear Fractional Maps

A composition operator C“, induced by a linear fractional map «,0 is compact if and
only if go(U) C U. Equivalently, C“, is not compact if and only if (p maps a point of
the unit circle onto the unit circle. This result can be deduced from the more general

theorem ([18], page 57):

Suppose (p is univalent self map of the disc. Then Clp is compact if and

only if (,0 does not have an angular derivative.

In this section we prove analogous results for the linear-fractionally induced op-
erators 0,20,), and 09°C,}.

The proofs will require both an explicit computation of the adjoint of a composi-
tion operator induced by a linear fractional self map of the disc, and some properties
of Toeplitz operators.

For g in L°°(6U), the Toeplitz operator T9 is the operator on H 2 given by
T9( f) = Pg f for f in H2, where P is the orthogonal projection of L2 onto H2.

(For more details on Toeplitz operators see [7] or [20]).

Cowen’s Adjoint Theorem([5], Theorem 2, page 153) Let 212(2) 2 (oz + b) / (cz +d)
be a linear fractional self map of U where ad—bc sé 0. Then 0(2) 2 (az—E)/(—hz+d)

maps U into itself, g(z) = (—52 + d)‘1 and h(z) = c2 + d are in H°°, and

Q=ndn.

In [5] Theorem 2, Cowen requires that ad - bc 2 1 but this is not necessary.

A comment about notation; throughout this chapter w and a will be linear frac-

tional self maps of the disc defined by the relationship,

;=ngn.

A calculation shows that w and a have the following nice properties:

C; = TEC‘IJT"? (1.1)

where 5(2) 2 (oz + d)’1 and h(z) = —liz + d,

if for some 17 E 8U, 111(7)) 2 w E 0U then o(w) : 77, (1.2)

and conversely,

if for some w E (9U, 0(a)) = 17 E 8U then 1,007) = w. (1.3)

Since CUT, = TfoaCa» for any f E H °° we obtain another expression for the adjoint

3=an up

where f(z) = g 0 04(2) 2 b2 + a.

A further comment about notation, by applying Cowen’s Adjoint Theorem and

. (1.1) we will represent the operators 0,7,, 0;, 0;, and 0% as:
,7, = TgCUTg, 0; = T50¢T§, 0; = TGCETI}, and 0;; = T50¢T§
Throughout this chapter we shall reserve the letters w, o, g, h, g, h, (p, Z, G, H, 0,

and H for this meaning.

Theorem 1.1 Suppose that go and 2,1) are linear fractional 3er maps of U. Then

0¢0,:, is not compact if and only if there exist points 771 and 172 E 6U such that
@071) = “’72) E 0U.

The key to the proof is the following lemma.

Lemma 1.2 Suppose that (p and w are linear fractional self maps of U. Then 0W0;

is compact if and only if 0¢00 is compact.

Proof of Lemma: Suppose first that 0¢00 is compact. Since

0,p0,';, = 0¢T900Tg (because 0,], = TQCUTg)

= TngwCan: (because 0¢Tg = Tgo¢0¢)

we see that 0¢0,7, is compact.

Conversely, suppose 0¢0,']‘, is compact. Since

CipCa' : CSP(C;)*
= csrxcyrg (by (1-1) and CJ=TECwTil

= Tzw0¢0$T§ (because CWT; = T30“) ,p).

it follows that 0.10,, is compact.

Proof of Theorem 1.1: By Lemma 1.2 the operator 0,,,0,}‘, is not compact if
and only if 0,,00 = 000,, is not compact. Since a o (p is a linear fractional self map of
U, 000,), is not compact if and only if a o (p maps a point of the unit circle onto the
unit circle. So there exist points 171 and n2 6 EU such that o o (p(7]1) = 172. Hence by

(1.3) there exists to E BU such that <p(n1) = w = M772), which completes the proof.

Theorem 1.3 Suppose that cp and w are linear fractional self maps of U. Then
0,],0,p is not compact if and only if there exist points wl and Log 6 3U such that

cp‘1(w1)= w‘1(w2) E 6U.

Proof: First we will reduce the compactness of 0,20,p to that of the composition
operator 000,, = Crow where 0,), = TQCUTA'. Suppose 0,20,p is compact.
First we will show that 0,20% is compact. By (1.4), 0,"3 can be written in the form

05 = 0¢T§TI§ where F(z) = 0 o go(z). Thus
0,20% = CleTFTitr

Thus 03,05 is compact.

Second we will show 000;; compact. We can apply Lemma 1.2 to conclude 0,7,0;
is compact if and only if 000,"3 is compact; in more detail: 0,7,0; = (020,1) is
compact if and only if (by Lemma 1.2) (0303' = 0005 is compact.

Finally we show 000,p is compact by observing that
000,p = 00(0,;)‘ = 00TH0§T5 = T300000§T5.

Thus 000,p is compact.
For the other direction suppose 0,,0,p is compact, and note that by Lemma 1.2
000,p is compact if and only if 000;; is compact. Now we will show that if 0009,, and

0,05 are both compact, then 0,130,, is compact.

By Cowen’s Adjoint Theorem, 0,7, = TQCUT; where h(z) = cz + d, thus

0,70,, = Tgcargo, .—_ marge, + argcac, (1.5)

By hypothesis 0,0,, is compact so the second term in expression (1.5) is compact. In
order to conclude 0,70,, is compact it suffices to show that the factor 0,T;0,, of the

first term in expression (1.5) is compact. The key to this is the following calculation:

cargo, = C,T;(C,7,)*
= C,T;THC';T5
= annoy;

= 00 Tb. 0;: TE;

_——-——

where b(z) 2 2H (z). By Cowen’s Adjoint Theorem we know that H (2) has the form

A2 + B. Thus b(z) = zH(z) E H °°, which implies that RC; = 0;Tb"o;. Hence,

C,T;C,, = c,c,gr;o,,rg,.

By hypothesis 0,0; is compact, hence 0,70,, is compact as desired.

Since cpoo is a linear fractional map of U, 0,0,, is not compact if and only if there
exist points cal and W2 on the unit circle such that <p(o(w2)) = wl. Thus there exists
17 6 EU such that <p(w1) = 17 and o(w2) = 17. By (1.3) w(17) = wz, which completes

the proof.

We will see that both of these linear fractional map results are prototypes for more
general theorems. Theorem 1.1 illustrates that the compactness of 0,0,7, depends on
the behavior of the range of (,0 and it. It illustrates the intuitive principle that if the

sets <p(U), ’l/J(U), and 0U are close then 0,0,7, is not compact. Similarly, Theorem 1.3

10

points to the fact that the compactness of 0,7,0, depends on the behavior in the
domain of (p and 112.

We now consider three examples that illustrate some differences between the com-
pactness of the operators 0,7,0, and 0,0,7. In all three examples cp and ib are linear
fractional self maps of the disc and the composition operators 0, and 0,, are not
compact.

Example (1) 0,7,0, is not compact but 0,0,7, is compact.

Consider the operators induced by,

1+2 —1—2
— 2 and 1,1)(2)— 2 .

 

 

«2(2)

The important points to notice are 90(1) 2 1 and 212(1) = —1. Since <p(1) 75 11)(1),
0,0,7, is not compact by Theorem 1.1. On the other hand, <p'1(1) = w‘1(—1), so
07,0, is compact by Theorem 1.3.

Example (2) 0,7,0, is compact but 0,0,7, is not compact.

Consider the operators induced by,

1+2 1—2
2 and w(2)— 2,

 

 

#90) =

where <p(1) = 1 and 21)(—1) = 1. Since <p(1) = ¢(—1), 0,0,7, is not compact by Theo-
rem 1.1. On the other hand, go‘1(1) # 112‘1(1), so 07,0, is compact by Theorem 1.3.

Example (3) Both 0,7,0, and 0,0,7, are compact.
Consider the operators induced by,
1+2 —1+2

cp(2)= 2 and 212(2):

 

 

11

where <p(1) = 1 and 1/J(—1) = —1. Since <p(1) 51$ i1)(-—1), 0,0,7, is compact by Theo-
rem 1.1. On the other hand, cp"1(1) 75 i1)‘1(1), so 0,7,0, is compact by Theorem 1.3.

1.1 Comparison principle.

We will now use Theorem 1.1, the linear fractional theorem for 0,0'“, to significantly
enlarge the set of operators 0,0,7 that we know are not compact. This is done in the

lemma below, but the key idea is the following theorem.

Theorem 1.4 (Comparison Principle for the Compactness of 0,0,7,.)

Suppose cp and w are univalent holomorphic self maps of U, and a and S are holomor-
phic self maps ofU such that a(U) C <,0(U) and fl(U) C t/)(U). If 0,0,7, is compact

then so is 0007;.

Proof: Because cp and 21) are univalent, and the range of (p and 't/) contains the

range of a and 6 respectively, we can form

71(2) 2 <p‘10(1(2), and

T2(2) = 111—1 0 5(3)

both of which take U holomorphically into itself. Thus, a(2) = (p o r1(2) and 6(2) =

11) 0 75(2), and at the operator level,

000;, = 0,0,0:

¢°T2

= 0,, 0,07%.

Hence 00,05 is compact.

Corollary 1.5 Suppose (,0 and 11) are univalent holomorphic self maps of U, and

12

<p(U) fl 212(U) contains a disc that is tangent to the unit circle. Then 0,0,7, is not

compact.

Proof: Let A be a disc that is contained in <p(U) fl w(U) and is tangent to the
unit circle. Then there exists a linear fractional map a that maps U onto A. Thus
a(U) C cp(U) fl 1,1)(U). Since 000; is not compact we conclude by the Comparison

Principle that 0,0,7, is not compact, which completes the proof.

Most theorems seem to come in pairs: one for the operator 0,0,7, and one for
the operator 0,70, but we are unable to find a comparison principle for the operator

0,70,.

CHAPTER 2

Preliminaries

In this reference section we introduce our notation, and sketch the prerequisites for

the rest of the paper.

2.1 An equivalent inner product on H 2.

The Hardy space H 2 is the collection of functions that are holomorphic in the unit
disc U and whose Taylor coefficients in the expansion about the origin are square

summable. H 2 is a Hilbert space where the inner product is defined by:

<f,g> = imam)

with f (n) and g(n) denoting the n—th Taylor coefficient of f and g respectively. The

Littlewood-Paley identity for the H 2 inner product is

<f,g>= f(0 g(0)+/U( f(2 WlogT— l dA(2) (2.1)

with dA representing normalized Lebesgue area measure, A(U) = 1. A calculation

with the Taylor series of f and g proves that these inner products are the same.

13

14

Moreover, when f = g we obtain the Littlewood-Paley identity for the H 2 norm

1112 = 1110112 +1, Hf’(z)|210g—LdA(2)- (2.2)

IZH2

2.2 Reproducing kernels and essential norm.

Let Ka(2) be the reproducing kernel at the point a E U and let k, be K, divided by

its norm,

 

 

2 -ar"’1/2
,a,z,_K.<>_<1 11)

_ HIKaHH _ 1—62 , (26U).

Let ”T”, denote the essential norm of the operator T on H 2 (recall that this is the
distance in the operator norm from the compact operators). Since k, converges to
zero uniformly on compact subsets of U as |a| —) 1‘ and ”k,” = 1 for all a E U, it

converges weakly to zero as |a| —> 1“. Thus, “Aka” —> 0 for every compact operator

A on H2, hence

“Tue 2 llT+A||
2 II(T+A)kaII-
Hence,

“Tue 2 lim SUP HHTk'aHH
|a|—+l

A result that follows immediately from the proof in [16] of Shapiro’s essential norm

 

formula is,
N a
lim sup ||0,lc,||2 = lim sup ’p( 1).
[aI—rl lal->1 03 W

Hence a composition operator’s action on the normalized reproducing kernels of H 2

15

completely determines the essential norm.

2.3 Nevanlinna counting function.

For a holomorphic self map (0 of the open disc U, we define the Nevanlinna counting

function of (p by:

N¢(w) = Z log-L, w E U\{<.0(0)}

zew—lhu) I I

where <p‘1(w) denotes the set of cp-preimages of w counting the multiplicity, and
N,(w) = 0 if w ¢ <p(U). One of the main ingredients in the proof of the essential

norm formula of a composition operator is the the following property of N,:

Sub-mean—value property ([16], Theorem 4.6, page 390) If A is a disk in U not

containing w(0), with center a, then

1
MWSWLMMMM

where [A] is normalized area measure of A.

The sub-mean-value property is used in [16] to establish the lower estimate,

N,(w)

0 2 Z limsup—.
Iwmlwrmfi

The next proposition follows directly from the proof of the lower estimate in [16]

and is a consequence of the counting function’s sub-mean-value property. Set

2r2|a|2 108 7,1,"
c,(a) = (1 + Hal) (1 _ Hal) and recall k, is the normalized reproducing kernel for the

point a E U.

 

16

Proposition 1 For 0 < r < 1 there exists 6 > 0 such that,

 

“0,13,,“2 2 c,(a) {3:02 for all 1— 6 g |a| g 1.
0|

Note that lim c,(a )2 r2.

Ia] [—+1—

2.4 Change of variables formula.

The following non-univalent change of variable formula was first used in the study of

composition operators by Shapiro in [16].

Theorem([16], Theorem 4.3, page 389) If F is a positive measurable function on U

and 1p is an holomorphic self map of U, then:

1,,1F212112'12Iog—AA12) )=2/1 F122 dAw1 ) 12.3)

The following calculation establishes the connection between composition opera-
tors and the Nevanlinna counting function. By applying the Littlewood-Paley

identity(2.2) for the H 2 norm to 0, f = f o (p we obtain,
1121.212 = 1,, 111221'12112102W2A12 21+ 11112101112

= [U 1,2 . ,(2121212112122 fiat/112 21+ 11112101112

: 21v 1 f’(w)l2N1p(IU)dA(’w) + 11112101112

where the last line follows from the change of variables formula (2.3), with g = | f’ |2.

Hence,

11 o 212 = 21,,11'1w112Nt12212A1w)+ 11112101112 f 2 H2. 124)

17
2.5 Littlewood’s Inequality.

Littlewood [8] in 1925 established the boundedness of composition operators on the
Hardy Spaces. The key to the proof is a result called Littlewood’s inequality, proofs

and development of this result can be found in [8], [16], and [18].

Littlewood’s Inequality.([16], Theorem 2.2, page 380) If 1,0 is a holomorphic self

map of the disc, then for each 2 E U\ {10(0)},

 

1 - 710(0)l
N 2 < lo —— . 2.5
In the case when 1,0(0) = 0, Littlewood’s inequality simplifies to
1
N,(2) 3 log — for 2 6 U \ {0}. (2.6)

IZH
An immediate observation from Littlewood’s inequality and the boundedness of
log |2| near 6U is that the Nevanlinna counting function is bounded near the boundary
of the unit disc. More precisely for each [10(0)] < r < 1 there exists a positive constant
0 such that

N,(2) S 0 for all |1p(0)| < r < 1. (2.7)

We now prove a lemma that we will use in the proof of Theorem 5.5.

Lemma 2.1 If 1,0 is a holomorphic self map of U, then

(1 - |r(0)l2).

[\DIH

1 Nt1212A1213
U

 

w . . . .
Proof: Set a,(w) = and notice that we can write Littlewood’s inequal-

l—wz’

18

ity as

 

N,(2) S 108

for all 2 E U \ 10(0).

 

 

01221010)

By applying successively Littlewood’s inequality, the change of variables w =
a,(0)(2), the Littlewood-Paley indentity (2.2) for the H 2 norm, we obtain the desired

result,

1

010(0) (2)

 

1U N,(2)dA(2) g 1U log 0121(2)

 

 

1 I 2
s 1, losWIOMOKwM 221122)

l I

1
= 5 (12,1011? — 12,2101?)

——- $(1—- 1.21012).

2.6 Angular derivative.

We say 1p has a finite angular derivative at a point C E 6U if there is a point 10 6 6U

such that the difference quotient

10(2) -- w
2 - C
has a finite limit as 2 tends non tangentially to C . The connection between composition
Operators and angular derivative depends heavily on the following classical theorem

of Julia and Caratheodory.

19

J ulia—Caratheodory Theorem.([18], Section 4.2, page 57) For C 6 EU, the follow-

ing conditions are equivalent:
1. 1p has a finite angular derivative at C.

2. 1,0 has a nontangential limit of modulus 1 at C, and the complex derivative 1p’

has a finite limit at C. In this case the limit of 112’ is 1,0’(C).

3. lim inf M

C 1 _ ,2, = d < 00. In this case, 1,0'(C) = 1p(C)Cd.

(For more information on the Julia-Caratheodory Theorem and its connection with
composition operators see [16], Section 3 or [18], Chapter 4)

The J ulia—Caratheodory Theorem allows us to think of |1p’| as a function mapping
the unit circle to (0, 00]. In the case when 10 is univalent it is shown in [3] that the
essential norm of 0, can be computed explicitly in terms of the angular derivative of
1,0. We reproduce part of the proof below. The argument relies on the fact that if 1,0

is univalent, then |1p’| is lower semicontinuous, a proof of which can be found in [3].

Theorem A ([3]) Suppose 1,0 is univalent. Then

—1

2 _ - I
HICAHI. — ggglsol

 

 

Proof: Applying Shapiro’s essential norm formula equation (1) of Chapter 1,
and noting that for univalent functions the Nevanlinna counting function simplifies
to log(1/|2|) where 2 = p'1(w) (with the understanding that log(1/|2|) is zero if w

is not in the image of 1p) we obtain

A
III

—1
1 — 1 —
”0,“: = lim sup 1 = lim sup——|Z|— = [lim inf M] .
[ml—)1— logm |z|—+l‘ 1" H(p(z)l |z|—)1" 1 _ HZ]

20

Upon applying the J ulia—Caratheodory Theorem to the term on the right, and noting

that by the lower semicontinuity |1,0’ I obtains its infimum on 0U, we see

—1
101.12 = [21111121] .

CEBU

CHAPTER 3

The Angular Derivative and

the Essential Norm

We now develop the link between the angular derivatives of 1,0 and 112 and the conjec-
tured essential norm formulas, equations (2) and (3) of the Introduction.

We use the following notation for nontangential approach regions: For 0 < p < 1,
let Ap(C) be the convex hull of the disc pU and the point C. For 0 < r < 1, let
Ap,,.(C) = A,(C) \rU. Let 1p*(C) denote the nontangential limit (when it exists) of
1,0(2). By Fatou’s Theorem this limit exists for a.e. C E 6U.

For to 6 EU we define

E(<p,w) = {C 6 6U. 10*(6) = w}

with the understanding that this set is empty if 10 is not a nontangential limiting

value of 1,0. Now we define for w E 8U:

1
6(90’w) = {Z HSp’(C)H ' CE E(901w)}1

 

where 1 / |1p’ (C )I = 0 if 1p does not have a finite angular derivative at C, and 15 (1;), w) = 0

if E(1p,w) is empty.

21

22

Our goal in this chapter is to generalize the following result;

Theorem ([16], Theorem 3.3) Suppose 1,0 is a holomorphic self map of U. Then

limsupfl’iu,32 Z sup {15(1,0,w): w E 3U}.

IwI-—>l’ 08 [7,7]

We will give two different generalizations of this result, the first of which is:

Theorem 3.1 Suppose that 1p and 11) are holomorphic self maps of the disc. Then

 

lim sup N,(w)N,,(2w) Z sup {6(1,0,w)6(11),w) : w E 0U}.

Ital->1“ (log IIVI)
Proof: Fix 0) 6 0U a nontangential limiting value of both 10 and 11). Suppose
{Ckfl’zl C E (1p,w) and {7737,21 C E (10,10) such that w has a finite angular derivative
at C1,, 1 g k g n and 10 has a finite angular derivative at 17k, 1 S k g m. Fix
0 < p < 1, and choose 0 < t < 1 so that the angular regions A,c = A,,t(Ck) are disjoint
for 1 g k g n, and similarly Bk = Ap,t(77k), 1 g k g m. Corollary 3.2 of [16] insures

that

(1112124.) = 1 s k s n}) n ((1111181) : 1 s k s m})

contains a nontangential approach region A with vertex 10.
For w E A \ {1p(0),11)(0)} choose a set of preimages of w for each inducing map 1,0

and 1,0, {2),(w)}7,‘:l and {uk(w)}7,":1, such that

1p(2k(w)) = w and 2,,(w) 6 A7,, 11: =1,...,n

111(uk(w)) = w and uk(w) E Bk, k = 1, ...,m.

23

By the definition of the Nevanlinna counting function we see:

 

2 Zn: log 1 and )>§:log (3.1)
k=1 sz(w)H k1: g]

uk(’w __)(1’wH.

For fixed k, we know by the Schwarz Lemma that 2,,(w) —-> C), and uk(w) ——> 771:
through A,c and Bk respectively as w —> 10 through A. Thus by the J ulia-Caratheodory

theorem:

log —1
H2100“ r —1

_ ¢ 302
10-111), wEA log rt, I (OCH ( )

10 0g luk(w)|
w—iw, wEA logI—-w I

=10 12111-1. (3.3)

Applying (3.1), (3.2), and (3.3) we obtain:

 

IV

1

N N z w m
limsup ,(w) ,(éw) limsup Z—i ’

lull—)1- (10g fi) 10-11;), 106.4,: _1 10g lwl k=l lOng—l

1 1

105521.122) m lim log T112)

__ —1

110—110, wEA log— W] 111—11», wEA log-I3,

 

k=l

 

 

= 211211 1112 101121.11

Now take the supremum over m and n and then the supremum over 1.0 E 6U to finish

the proof.

An immediate consequence of the proof of Theorem 3.1 is;

Corollary 3.2 If there exist three points C, 17, and w on the unit circle such that

10(C) = 1(0) = w and 101C) and 11/01) exist, then

- N2(w)NA(W) 1
11m sup 2 Z —7—,.
""W (log Iwil) H‘p (Clip (77)]

 

24

Our second generalization is:

Theorem 3.3 Suppose that 1,0 and 112 are holomorphic self maps of the disc. Then

N2(<p(2))N1/2(w(2))
|z|—)1' log m log lip—(1,7)]

 

2 811130500, ¢(C))5(¢,¢(C)) : C E 0U}.

Proof: Fix C E 6U such that both 1p and 10 have a finite angular derivative at
C. Let 101 = 1p(C) and 102 = 112(C) be the nontangential limiting values of 1,0 and 11) as
2 —> C nontangentially. Suppose {Ck}2=1 C E (1p,w1) and {17),}7;1 C E (10,102) are such
that 1p has a finite angular derivative at C7,, 1 S k S n, and 1,0 has a finite angular
derivative at 17k, 1 S k S m.

Fix 0 < p < 1, and choose 0 < t < 1 so that the angular regions A), = Ap,t(Ck) are
disjoint for 1 S k S n, and similarly for 8,, = Ap,t(17k), 1 S k S m. Corollary 3.2 of

[16] insures that the set

(HM/412) = 1 S k S 11}

contains a nontangential approach region A, with vertex 1.01, and that the set

11111181.) : 1512:2221

contains a nontangential approach region B, with vertex 102. For a point 2 E U such
that 1p(2) E A, and 10(2) 6 3,, choose a set of 1p-preimages {12),(1,0(2))}7,‘=1 for 1p, and

choose a set of w-preimages {uk(11)(2))}7,”:1 of 10(2), so that

1001120112)» = 10(2) and vk(<p(2)) 6 A12, k = 11m,”

111(uk(112(2))) 2 10(2) and uk(w(2)) 6 B7,, 11: = 1, ...,m.

25

By the definition of the Nevanlinna counting function we see:

 

n 1 m
221721.012» and N11211az1§wg5—,—,,z,, (3'4)

For fixed 11:, we know by Schwarz Lemma that vk(10(2)) -—) Ck and uk(111(2)) —1 17,c
through A,c and B), respectively as 2 -—> C through A = 1p‘1(A,) fl 1,0-1(B,). Thus by

the Julia-Caratheodory theorem:

 

 

l—g'——‘—— —'11121 «11 1 13.51
“C 16” 10g 121211

log——
1' M: ' -1, 35

Applying (3.4), (3.5), and (3.6) we obtain:

N N z gu z
limsup lv(¢(flllw(¢1(2)) 2 1,1118“,sz 1121.121):_°__1 11.1121
12"” 0g1721211 °g_1212121"1 HUG/12-1 081,01.“ 1._ 081.1(2)]
m

___1_ _1__
_ :Iim log vk(<P(z)) lim 10 OgUk(¢(Z)) (Z 6 A)
1“”EA 1031121211 12: 1W6” 10g1121211

 

 

Now take the supremum over 11 and m, and then the supremum over C 6 EU to finish

the proof.

An immediate corollary of the proof of Theorem 3.3 is

Corollary 3.4 If 1,0 and '11) have an angular derivative at the point C E 0U then

 

. N 12121111111121) 1
1 ‘P > -———.
‘Tfé‘p lam-.2212 - 11211111111

CHAPTER 4

The Operator 0,70, on H 2

This chapter is broken up into three parts: In the first section we outline the main
results for the operator 0,70,, and using Theorem 3.3 we develop the connection
between the compactness of 0,70, and the angular derivative of the inducing maps.
In the second section we establish an upper bound on the essential norm of 0,70,, and

in the third section we prove a necessary condition for the operator to be compact.

4.1 Main results for 0,7,0 .

Theorem 4.1 Suppose that 1p and 10 are holomorphic self maps of U. Then

 

N.1121211N11211211) ”2,

“0,70,“, S lim sup ( 1 1
log 112—1211 10g _111211

I2I-11‘

Corollary 4.2 Suppose that 1p and 11) are holomorphic self maps of U, and

lim sup1\I1p(<p(f))IV¢(1/2Iz)) = 0.
lzlrl‘ 0g IBM 03 “‘111‘211

 

Then 0,7,0, is compact on H2.

26

27

Remark. In the case when 111 = 1p, Shapiro’s essential norm formula tells us that,

N,
“0,7 CWHHe — IIC1pII2— — lim sup 1M
||z|—+1-OgI,p,zI

hence the estimate of Theorem 4.1 is sharp. We now consider the form that The-
orem 4.1 takes when the inducing maps are univalent. If 1p is univalent then
the Nevanlinna counting function of 1p, evaluated at 10(2), simplifies to 1081-17], i.e.
N,(1,0(2)) = log lit—I' Thus if 1p and 10 are univalent the above upper bound on the

essential norm simplifies to;

 

 

 

2
, N 2 N 2 . log;
limsup 159021;wa )) — limsup10 —(l—llol) 1
121—21- 81.1.11 8—10.11 Izl-H‘ g121211 g1111211
_ 2
= limsup (1 '2')

121—21— (1 - |<P(z)l)(1 - |¢(2)H)'
Hence an immediate corollary of Theorem 4.1 is;

Corollary 4.3 Suppose 1,0 and 11) are univalent self maps of the disc, and

. 11 — 12112 _
12’15’3- 11 — 1121211111 — 12112111 ’ 0'

 

Then 0,70, is compact.

The next theorem is stronger than the converse of Corollary 4.3 in that we only

assume one of the inducing maps is univalent.

Theorem 4.4 Suppose 1p and 11) are self maps of the disc and one of 1p or 11) is

univalent. If 0,7,0, is compact on H 2 then

. 11 -— 12112 _
1315’?— 11- 121211111 — 11112111 ’ 0'

 

28

By Theorem 3.3 this says that if 0,70, is compact then 1,0 and 10 do not have an
angular derivative at the same point.

Putting together Theorem 4.1 and Theorem 4.4 we obtain the following com-
plete function theoretic characterization of compactness when the inducing maps are

univalent.

Corollary 4.5 Suppose 1p and 11) are univalent self maps of the disc. Then 0,70, is

compact on H 2 if and only if

. 11 — 12112 _
1115’?— 11— 1121211111— 12112111 ‘ 0'

 

In terms of the angular derivative of 1p and 111 we obtain the equivalent statement:

Corollary 4.6 Suppose 1p and 11) are univalent self maps of the disc. Then 0,70, is

not compact if and only if 1p and 1,!) have an angular derivative at the same point.

To connect Corollaries 4.5 and 4.6 with the work we did in Chapter 1, note that if
10 is a linear fractional self map of the disc then the existence of the angular derivative
is equivalent to 1p mapping a point of the unit circle onto the unit circle. Thus if 1p and
111 are linear fractional self maps of U, Corollary 4.6 implies that 0,70, is not compact
if and only if there exists a point 77 E 6U such that 1p(17) E EU and 10(17) E 6U.
Equivalently, there exist points 101 and 102 E 0U such that 1,0‘1(w1) = 111‘1(w2) E 0U,
which is Theorem 1.3.

We now turn our attention to the proofs of the theorems.

4.2 Upper estimate on the essential norm of 0,70 .

The proof of Theorem 4.1 is based on the approach Shapiro used in [16] to find the

upper estimate of the essential norm of a composition operator. We will use the

29

following general formula for the essential norm of a linear operator on a Hilbert

space.

Proposition 2 Suppose T is a bounded linear operator on a Hilbert space H. Let
{Kn} be a sequence of compact self-adjoint operators on H, and write RR = I — Kn.
Suppose ”Ru” 2 1 for each n, and ”Ram” —) 0 for each a: E H. Then ”THC =
limn lanTRnH°

The proof of Proposition 2 follows directly from the proof of Proposition 5.1 in [16].
We also require the following estimates on H 2 functions, whose proof can be found

in [16].

Proposition 3 Suppose f E H 2 has a zero of order at least n at the origin. Then

for each z E U:
1. |f(Z)| s lzl"(1- Izlz)‘%||f||2, and
2. lf’(z)| s finlzl""‘(1—Izl2)‘%||f||2.

Proof of Theorem 4.1: We will show that

 

”012ch le S lim SUP

IzI—il

(Ni<¢1z))Ni1¢(z)))l/2.

1 1
log 1712» 10g “—1sz

This will be done by applying Proposition 2 with Kn the operator that takes f to the

nth partial sum of its Taylor series:

an(z) = i f(k)zk , where f(z) = f: f(k)zk 6 H2.
11:0 12:0

Since Kn is the orthogonal projection of H 2 onto the closed subspace spanned by
the monomials 1, z, ..., 2", it is self-adjoint and compact. Since Rn = I — Kn is the

complementary projection, its norm is 1. Thus the hypotheses of Proposition 2 are

30

fulfilled, so that

“Czlzcvlle S lifPHRnCJJCan“ = lifrln SUP l < CchnfaCt/IRng > I (4-1)

1996(H2)l

where (H 2)1 is the unit ball of H 2. To estimate the inner product on the right hand
side of equation (4.1) it is enough to consider the supremum over the unit ball of H3
because Rn f and Rug are in the unit ball of Hg for all n 2 1 and for all f and g in
(H 2)1. Therefore, to estimate the inner product, fix the functions f and g in the unit
ball of Hg, and a positive integer n, and use the Littlewood-Paley identity for the H 2

inner product to obtain,

I < CoRflfflwRag > | S |R«f(90(0))Rn9(1/2(0))l (4-2)

+ [U |(C,R.fy(z)<ciag>'1z>llog—Lame). 14.3)

IZI2

Since R, f and Rug are in the H 2 unit ball and both have a zero of order n at the

origin, Proposition 3 implies that

 

lam/210)): s |so(0)I"/\/(1-|10(0)l2), 14.4)
and
|(R«.f)’(z)l2 s 277u2lzl‘*"“”/(1-Iz|2)3. (4.5)

and similarly for |Rng(1p(0))| and |(R,,g)’(z)|. Now fix 0 < r < 1 and split the integral

on the right side of equation (4.3) into two parts: one over the disc rU and the other

31

over its complement. Use estimate (4.4) on the first term of equation (4.3) to obtain,

 

lcp(0)|"|1/2(0)|"
|<C,,R,,f,CR.. >| <
"’ g 1/(1-l10(0)l2)(1-I1/2(0)l2)

+ frU|(C.pR..f)’(z)(C¢Rng)’Z( 2:)IlogI— I dA12) (4.6)
I I 1
+ (U122 110.12.!) (swim) (2)! log l35112112)-

The first term on the right hand side has limit zero as n tends to infinity, so we need
only be concerned with the two integrals. Let I represent the integral over rU. Set
p = sup {max(|1p(z)|, |1/2(z)|) : z 6 rU}, which is clearly less than one. To estimate
I, use the Cauchy-Schwartz inequality, the change of variables formula (2.3), and

estimate (4.5) to obtain

I = l. UI1C.R.f>'12>1cwng)'12)llog WOW 2)

s ([ul1cian'1znzlogI—1I-2-dA12) ([1 I1'20iR.g>1>IZIogI—— —I—,dA12))l/2

1/2

s 2 (/ 11R.f)'1w)I2N:1w>dA1w))”2 (fU11R.g)'1w)I2Ni1w)dA1w))

S [tn—Lh— 1) )3(/U N¢( A(w))l/2 (fl) N¢(w)dA(w))l/2
2(n—l)
3 2,1112;

The last inequality follows from Littlewood’s inequality (Lemma 2.1, Section 2.5).
Thus the supremum over f and g in the unit ball of Hg of the integral I is bounded
by an expression whose limit is zero as n tends to infinity.

A note about notation; an unadorned ” sup” will mean the supremum over f and

g in the unit ball of Hg throughout this section.

32

We have

1 2 2(n—1)
—dA(z) < 2” ’0

sup [UI1C.R.f)'12)1CiR.g)'12)IlogIZI, 41—1173)? 14.7)

Therefore

1111.11.20.11.” s sup [WU |(02Rnf)'(2)(C¢Rng)’(Z)l logI—IgdA12 2)

2n2p21n_u+ I<p(0)l"|1/2(0)l" ,
"11- 22)3 ,/11— 11210)12)11 - 11210)?)

 

 

We may replace R, f by f and Rug by g in the above integral because f and 9 range

over a larger set than their projections R, f and Rug. Hence,

1 211.4(2)

”11.11.20.211.“ 3 sup [\Ul 2)10.)g)12 z)|logI— I

2n2p2("_1)(m+ mammal" .
11 — W1 1/11 — lr(0)l2)(1 — 111W)

Now let n tend to infinity. Because p, |1p(0)|, and |1p(0)| are less than one, we conclude

 

“05.0.”. s sup [U 1.2 |(f o 2)'12)1g o w)'12)110g $112112). 14.8)
To finish the proof set h,p(z)=‘1]—VLZ) and
03121
1/2
Hz 2 h... 2 h 1122 ./2=(N.1212))N21¢12))) _
1) 1 1121)).111)» IOEWIOEW.»

We have

||0i012||2< _ sup/I1)'f0121gow)'llongA12)

33

 

= 2sup/IUI l)’(foso g)1020)'IH1)H(——I—Z-:—100I IdA12)
s 2r<sI1:Ip<1{H(Z)}8upu mu |(foso)’(g°1/1)’| —(—Hz)logI:—IdA(Z)
|(f<><p)(z)’||(’.11<>111)(2)|l
u H(z u 1 , —dA 2
“(2312135 }S p[1101.11012))01h.1012)))1 °glzl ()
1/2
su 8111:?sz o 2
S 2T<I2Ip<1{H(z (25)} p(/U|(f01p) —(_1p(‘Pz))1glzlldA( ))

((UK .1..)I2m10 $112412) 10
9° 111.1012» ongl ’

where the last line follows from the Cauchy-Schwarz inequality. Now we will calculate
the two integrals in the last expression above; because the calculations are similar we
will only explicitly compute the first integral. To do this, use the change of variables

formula (2.3) and the Littlewood-Paley identity (2.2) for the norm to obtain ,

o zolgIT‘lzn 0 i
[Um 10)'I N.1 ——Iz—IIIgIZIdA12)

)2 2101ng 1 _
=/ lf’zo(<p l)|<p(z)l N.“1_"1())')‘°gI?IdA‘z) 101 00—1012)

— —/1 lf’(w I2—g——,3""N.w1 )0A1w)

— —/1 lf’('w w)I'100—IdA1w 10)

= 5M“?-

Similarly,

log —1 1 1
I 2 |¢(Z)| = _ 2
l. I10011) 12)I —N 100 dA12) 2IIgII .

1110/42» |7|

34

Since ”f“ = ||9|| = 1. we get
”Cbcsplle S SUP{H(2) 3 7' _<. M <1}:

and the desired result follows upon letting r tend to 1‘.

4.3 Necessary condition for 0,20.p to be compact.

We will prove the contrapositive form of Theorem 4.4:

Suppose 1p or 1,1) is univalent. If 1p and 11) have a finite angular derivative

at the same point of EU then 0,20,p is not compact on H 2.

Suppose 1p is univalent and let 17 E 6U be such that 1p’(77) and 111' (17) exist. We may

assume the following;

(1) n = 1 (by rotations)

(2) 1p(1) = 1 (by rotations)

(3) 1p’ (1) = 1 (by hyperbolic automorphism)
(4) ||0¢||e = 1 (by parabolic non-automorphism)

The first three modifications above will be obtained by multiplying 0,)",0Ip by an
invertible composition operator or by the adjoint of an invertible composition operator
thus not changing the compactness of 0;,0Ip. The fourth modification will involve
multiplying 0,“,‘0.p by a non-invertible composition operator, this is not a problem
because if the product of 0,7,0,p with any operator is not compact then 0,7,0Ip can not
be compact.

To see why we may assume conditions (1) and (2), let a be a rotation of the

disc that takes the point 1 to 77. The induced composition operator is unitary, i.e.

35

051 = 0;. Let S be the rotation of the disc that takes 17 to 1, note fl = 0*. Then
0,};0,p is not compact if and only if 0,203,000.35 = 0,200,030.,” is not compact. Since
both 1b 0 oz and B 0 1p 0 a have angular derivatives at 1 and [3 0 1p 0 a(1) = 1 we will
assume 7) =1 and 1p(1) = 1.

To see why we may assume condition (3), we know by (1) and (2) that 1 is a
boundary fixed point of 1p, so 3 = 1p’ (1) > 0. Let r be a hyperbolic automorphism of
the unit disc that has fixed point, with r’(1) = l/s. Then (7'01p)'(1) = 1, ro1p(1) = 1,
it suffices to show 0,130....p = 0,7,04,07 is not compact inorder to conclude 0,7,0.p is not
compact. Thus we may assume that 10’ (1) = 1 .

To show that we may assume (4), recall from Theorem A in Section 2.6, that the

essential norm of a univalently induced composition operator 0Ip is given by,

||0Ip||§ = max{ ' 77 E 8U}. (4.9)

1
W '
Let S be a linear fractional self map of the unit disc, not an automorphism, with 1 its
only fixed point, i.e. fl is a parabolic non-automorphism of the disc with fixed point
1. It can be shown that the derivative of fl at 1 is one, thus the angular derivative is
one. Also, since fl is a non-automorphism, 1 is the only point for which the angular
derivative exists. So fl 0 1p has boundary fixed point 1, angular derivative one at
1, and 1 is the only point for which the angular derivative of S 0 1p exists. This
implies, by (4.9), that the essential norm of 030., is one. It suffices to show that
0,7,0)“Ip = 0,20,,03 is not compact, and 05,050,, is an operator with fl 0 1p having all
the desired conditions (1),(2),(3), and (4). Thus we may assume that ||0Ip||e = 1.

We continue the proof of Theorem 4.4 under the assumptions (1)-(4) on the uni-
valent map 1p. Consider the family of normalized reproducing kernels {k,(z)} for

0 < r < 1, where
Kr(z) _ \/1-— r2
“K,” — l—rz '

 

 

kr(z) =

36

Since {kr} converges weakly to zero as r —> 1, it will suffice to show that
lim sup ||0,‘I',0,pk,.|| > 0, (4.10)
r—->1

thus showing 0,20,p is not compact.

Set g,. = 0,pk,. - k,, which implies

“0.2012112” = llCikr+Cigrll

“€5.er - llelngrll- (4011)

IV

We will now show lim,_,1 ||0,‘I‘,k,.|| > O and lim,_,1 Hg,“ 2 0, which will prove inequal-
ity (4.10).
Since 0,;Kw 2 KW”) and 112’ (1) exists, upon applying the Julia-Caratheodory

Theorem we obtain,

 

gig; “Ci/0.“? = 113011— roux/.1.“
— lim 1—r2
“ r011 — 1012)I2
— lim 1—r
H1 1 - |¢(r)|
_ l
lp'(1)|
> 0
Thus lim,_,1 ||0,j,k,|| = I WlUl > 0, so we have reduced the problem to showing that
1imr—01llgrll = 0-
llgrll2 = ||010l€r—krll2

< KT 0 (p) Kr >
llKrll2

 

= IIkarll2 + llkrll2 - 236

37

1—r2

= 10,2 1—2R .
IIC. II + 81—21012)

(4.12)

The lim sup as r —) 1 of the first term is bounded by the essential norm of the operator

0.p which by hypothesis is one, i.e.
lim llekrll2 S lim SUP llapkwll2 S “00“.? = 1-
T—)1 I‘wI—H.

Now to finish the proof we have to deal with the third term in equation (4.12), which

we do in the following calculation,

 

 

1 - 790(7") _ 1 (1 - W‘) + W“) - WW)

 

l—r2 _ 1+r l—r
1 1-10(7‘)
1+r(1—r +1p(r)).

And since lim,_,1 M52 = 1p'(1) = 1 and 1p(1) = 1, we conclude that

l—r

Hence lim,_,1 “g,“ = O, which completes the proof of Theorem 4.4.

CHAPTER 5

The Operator 0Ip0,’z on H 2

This chapter is broken up into three parts: In the first section we outline the main
results for the operator 0,,0,’;, and using Theorem 3.1 we develop the connection
between the compactness of the operator 0.,05, and the angular derivative of the
inducing maps. In the second section we establish a lower bound on the essential
norm of 0,,,0" , and in the third section we prove a sufficient condition for the operator

to be compact.

5.1 Main results for 09.05,.

We establish the following lower estimates on the essential norm, which provides a

necessary condition for 0,,0; to be compact.
Theorem 5.1 Suppose 1p and 11) are holomorphic self maps of the disc. Then

10 .1.
MIICCJII2>limSUP10gIzI “0&2”, and
121—11 031112) 0g 1012))

logi N
2 no, 0,7,“: >nmsu_pl IzI 1009(2))
'2 *1 0gI1012)I 0gI1.012)I

 

 

Remark. In the special case when one of the inducing maps is univalent Theo-

rem 5.1 reduces to the lower estimate on the essential norm of 0,00,”; conjectured in

38

39

equation (3) of the Introduction. To see this suppose v is univalent, and apply the

change of variables 11) = 112(2) to the first lower estimate in Theorem 5.1 to obtain,

 

“0.0;“: 2 11m 000 Nr‘w)N"(§“).
IwI—n- (log F171)

With a little more care we obtain the following,
Corollary 5.2 Suppose 1p and 2/2 are holomorphic self maps of the disc and either 10

or 1b is boundedly valent. Then

||0¢CQII§ Z MlimsupN¢(w)N‘p(:U). (5.1)
le-H‘ (logI—th)

 

where M is a positive constant.

The next corollary is a sufficient condition for noncompactness of 0¢0,j, in terms
of the angular derivatives of the inducing maps 1p and 1/2 and is a generalization of the
angular derivative criterion for a composition operator. The corollary follows directly

from Theorem 5.1 and Theorem 3.1.

Corollary 5.3 Suppose (1, (2, and w are three points on the unit circle such that

1. 1p(C1) = 1/J(C2) = w and.

2. 1p’((1) and 1mg.) exist.
Then 0,00,}, is not compact.

The two lower estimates on the essential norm in Theorem 5.1 are not sufficient for
compactness of a composition operator. This can by seen by considering the inducing
map 10 defined in [18] page 185: an inner function which does not have an angular

derivative at any point. By the Julia-Caratheodory Theorem the non-existence of an

40
angular derivative is equivalent to,

logIZI

lim sup 2 O,

IzI—n- log I<P(Z)I

and since 1p is inner, its counting function approaches zero slowly in the sense

lim sup M

> 0.
|w|—+l‘ logfifi

The next theorem is a sufficient condition for 0.0,}; to be compact. Roughly the
sufficient condition says that if 1p(U), 1p(U), and 3U are not too close then 0.,,0,‘,‘, is

compact. We use the following notation,

 

, N
EIP = {C E 6U : lug—1:21p log((%)) > 0}. (5.2)

Theorem 5.4 Suppose 10 and «p are holomorphic self maps of the disc. If

dist(EIp, E¢) > 0 then 0Ip0,'I‘, is compact.

It is a straightforward calculation to show dist(E,p, E10) > 0 implies

lim sup N,p(w)N,/,(;v) = 0.
IwI-H‘ (log Till—I)

 

Thus Theorem 5.4 is a partial converse to Theorem 5.1.

5.2 Lower estimate for the essential norm of 0I00,’I",.

Theorem 5.1 Suppose 1p and 1b are holomorphic self maps of the disc. Then

Iogi N
1.1—m og W1.» 03 'I0‘10)I

10gi N
2. ||0¢C§2IIZ Z limsupl “11' 1¢(90(la))_
IaI-+1‘ Og—l1p(a)| 0910—10))

 

 

41

Proof: Let Ka(z) be the reproducing kernel at the point a E U and let ka be

Kc divided by its norm,

K.12) _ 11 - 10W”
IIKaII _ 1 — 6’2

 

 

10(2) = , (2 e U). (5.3)

Since ka converges to zero uniformly on compact subsets of U as |a| —> 1' and

Ilka” = 1, it converges weakly to zero as |a| —> 1". Hence,
||0,,,0,}‘,||,.3 _>_ lim sup ||0Ip0,';,ka||.
|a|—)1-

Using the identity 03K“ 2 K1101) and normalizing KW” we obtain,

1 - |a|2

0Ip0*k=a21—a 0¢K02= ——
II II 1 mu .1)“ —I010)I2

I|C10kw1a)ll2-

Therefore,
1

logI—I
limslup||0,p 0,,ka H2- — limsup10g II“ ”Ca l€11)(a)II2
||a—>1- gIt‘ll—na—N

 

Now fix 0 < r < 1. By Proposition 1 of Section 2.3 we obtain,

N 01(1P(a))

“Cw kw: )II2> _
I0 OgI—wlan

cr(a)

for 1p(a) sufficiently close to EU. Thus,

 

log-I- N
limsup||0,p 0" k0,”2 > limsupc,(a ) Ill rU./1(3))
IaI—*1|a|->1-10g W 10g W

1
= r 2limsup IOgmN(1p(1a))
|a|-+1“ 108—“ 1.1.1.» 103— lean

 

Since r can be chosen arbitrarly close to one, this completes the proof for the first

lower estimate.

42

Applying the above calculation to the adjoint of CIPC‘, which is 0,005, we obtain

the second lower estimate, thus finishing the proof.

Corollary 5.2 Suppose either 1;) or 1,!) is boundedly valent. Then

||C',p C"‘||2 > Mlimsup N‘p(w)Nw(;U)
IwI—n- (log I37)

 

where M is a positive constant.

Proof: Assume 1/1 is boundedly valent. Let 190(2) 2 ha(z) / Ilka“ be the normalized

reproducing kernel at the point a E U. For 11) E U set {oi}? = 1Z1'1(w), and

Nil»-

71

Fw,w(2) : Zcikai(z)1 where Ci = (Z: IIKaj “—2) IIKmII—l-
1:1

i=1

Since 11; is boundedly valent IIF¢,wII is uniformly bounded for all 111 E U and Fwy con-
verges to zero uniformly on compact subsets of U as |w| —> 1‘. Thus Fwy converges '
weakly to zero as |w| —> 1’. Let l/M be an upper bound on IIF¢,wII for all 111 E U.

Hence

 

 

 

C‘F w
”Cg, CIZIIe Z limsup ”0 9p ‘0 .11, H > MlimsuflpllC C¢F¢w||. (5.4)
Ital—>1" ”Fwy” |w| —11
Using the fact that CtzKai = Kw we see
.. " c:- . " cz-
C¢Fw.w(z) = Z “k ”Cszai(z) :. KW(z) Z ”k I ' (5'5)
i=1 01' i=1 at

 

A short calculation shows

 

: “KL.” =(Z IIK:_T_II2)1/2' (5.6)

43

Thus using equation (5.5), equation (5.6), and normalizing Kw, we obtain,

no 0"le = OK",
99 1.0 w I II IIZIITZ,”

 

= ”C K‘”” (.2 ”Kind:

:1

 

IIKwII2
= IIC k II (:1
Z llKa.||2
Therefore, by equation (5.4) and Proposition 1 of Section 2.3 we obtain,

”01.0.3.1. 2 Mlimsu_pllC C ...FIwII

|w|1—)"

. N1w) I__lelP
> 1 “’
— 1‘31??? (I__-cg 1.)”? (23— IIK..II2

1/2
2 limsup (Ni1wINI1wI)

 

|w|—)1' 10g I—it—I 10g Till—I

thus finishing the proof.

5.3 Sufficient condition for 0100;), to be compact.

This section is broken into three parts: We start with a technical lemma. We then
prove a theorem that is weaker than Theorem 5.4 and, using this result, we prove

Theorem 5.4.

Lemma 5.5 Suppose w is a holomorphic self map of the disc and f E H 2 of norm

one. Then

2 11210111
|(1Pf 2“ <4/I)N |1—(zw)|5d A(w>+I1—7¢(0)I4.

 

44

Proof: Let Fz(w) be the reproducing kernel for the derivative of an H 2 function

at the point z E U, i.e.

f'(Z)=<f,Fz> forall feHz.

In particular,

(03f) (z) =< o; f, F. >=< f, F, 0 II, > . (5.7)

Fix f E H 2 such that M f H = 1. Then use the inner product representation
in equation (5.7) to estimate | ( ,Zf)’ (z)|. Apply successively, the Cauchy-Schwartz

inequality, the fact that M f H 2 1, and the change of variables formula (2.4), to obtain:

KCJZfl'IZII2 = I < LE. 010 > I2
S IIf|I2IIFz 011)“2
: ”Fr. 0 1i)”2

= 2 [11:10) IF;(w)|2N,(w)dA(w) + IF. 0 «MOM2

|/\

4/I——N"’(w)1m(w)+|F 01210 )l

1 - 211) I6
which is the desired result.
Theorem 5.6 If dist(E,p,1/J(U)) > 0 then CIpCIj, is compact.

Proof: Let (fn) be a sequence in H 2 that converges uniformly on compact sub—

sets of U to zero and || nt| = 1 for all n. Thus (fn) converges weakly to zero. We will

45

Show,
"131010 IIchhntI = 0-

To estimate ||C¢C,7,fn||, use the Littlewood-Paley identity(2.2) for the H 2 norm

and the change of variable formula (2.4) to obtain,
IIC.C.7.f.II2 = 2 [U I1cif.)'I2N.dA + Icicifnwm

Since (fn) converges to zero weakly and CIpCJ, is bounded operator, (C¢C,2,fn) con-
verges to zero weakly. Thus IC1pCJ, fn(0)| converges to zero as n —-> 00. Hence as

n —> 00 we obtain,
Inc/“.21.“? = 2 [U I1Cif.)'I2N.dA + 011). 15.8)

Now temporarily fix 0 < r < 1, and split the integral on the right side of (5.8) into
two parts: one over the disc rU, and the other over its complement. Since C;f,,(z)
converges weakly to zero it follows that |(C,7,fn)’(z)| converges uniformly to zero on

the relatively compact set rU as n —> 00. Thus we obtain,

It 2 _ 10: I2
IICIpCian — 2 UIrU+2/rul( .f.)IN..dA+o11)

= 2 *n’szA 1
U\TUI(C¢f)I «p +0()

as n —> 00. We have reduced the proof to estimating the integral,

_ t I 2
I... — [U W 110.1.) I Ind/112).

46

More precisely, we need to Show that given 6 > 0 there exists an O < r < 1 and a

positive integer N such that

IrnSe forall nZN.

1

Since dist(E,p,1/2(U)) > 0 (where E,,, is defined by equation (5.2)) there exists a

subset S of the unit disc such that

E10 C S and, (5.9)

dist(S, $111)) > o. (5.10)

Now split the integral I”, into two integrals: one over S, = S D (U \ rU) and the

other over SC 0 (U \ rU) to obtain,

1,,,=/+ C'n’2NdA.
, f MW)" ..m i

Let 6 > 0. We will estimate each integral separately starting with the integral
over 3,. Choose 0 < r < 1 so that 1,0(0) 9! U \ rU and A(S,.) < e and let C be a
constant such that

sup N,p(z) = C.

r§|z|<1

Upon applying Lemma (5.5) we obtain,

[3. I1cifn)'I2N..dA1z>

 

 

g 4 f . [I (U) NI”1(13);Y:I(ISZ)dA(w)dA(z)+ s. |1 ivgfoihn‘id‘flz)‘ (5.11)

47

Since dist(S,1/)(U)) > 0 and S, C S it is clear that dist(S,.,1p(U)) > 0. Thus there

exists 6 > 0 such that
inf{|1— mIG: z e 5., w e 112(U)} > 6. (5.12)

In particular |1 — 71/)(0)|4 > 6 for all z 6 Sr. Hence we estimate the first term on
the right side of expression (5.11) by applying successively inequality (5.12) and

Lemma 2.1 of Section 2.5 to obtain,

 

 

N (w)N <,,() 1
[T/(U) Ibl -Iz'w|6 dA(w )dA(Zl S Eff/UNIp(z)N¢(w)dA(w)dA(z)
S A?» squ NIp(z) N¢(w)dA(w)
rglz <1 U
s (—,%)11—I¢10>I2).

We estimate the second term of expression (5.11) to obtain,

 

 

NIp(z) 1
(1—22110)I4dA(Z) s 6 SN..z1)dA1z>
C
S 69-.

Hence the integral over S, is less than a constant multiple of c, with the constant
independent of n.
We now consider the integral over SC 0 (U \ rU), i e

C“ n ’2N dA.
[WW I1 .1 )l 5

48

Since E90 C S we see from the definition of Ev, equation (5.2), that

lim sup M = 0.

1
(2141,2631: 108 |7|

Thus there exists 0 < r < 1 such that

NIp(z) g elogIéI for all r 3 [2| <1 and z 6 SC. (5.13)

Hence applying inequality (5.13) and the Littlewood-Paley identity (2.2) we obtain,

|(thn)’(2)l2N.o(z)dA(z) s eLI1Cif.)'1z)I2logI§dA1z>

§(IIC.;II2 — I1 5151012).

[9CD(U\rU)

|/\

Thus the integrals over Sr and SC 0 (U \ rU) are both less then a constant multiple

of 6, with each constant independent of n. This finishes the proof.

Before starting the proof of Theorem 5.4 we introduce the definition of a smooth

sector. First by a sector we mean the interior of an angle with center at the origin.

Definition: A subset S of the unit disc is a smooth sector if S is contained in a sector
of the unit disc and the boundary of S is smooth in the following sense: let T be a

Riemann map from U to S, then

lim inf N,(2)

1 > 0.
|z|—)1‘,z€S log I?!

 

Theorem 5.4 If dist(E,p, E¢) > 6 > 0 then CIPCJ, is compact.
Proof: Without loss of generality we may assume that 90(0) = 0. Let (fn) be

a sequence in H 2 that converges weakly to zero. Using the same argument as in the

49

proof of Theorem 5.6 we reduce this proof to estimating the integral,

1,, = [51.5 I1 .an)'l2N<pdA(Z)-

More precisely, we need to show that given 6 > 0 there exists an O < r < 1 and a

positive integer N such that
I”, S e for all n 2 N.

Let 6 > O.
For each point C 6 EW let [C be an arc of the unit circle with center C and arc
length 6 / 2. By hypothesis dist(E,,,, E,/,) > 6 and since the arc length of I; is 6 / 2, it is

clear that

dist(I(, Ew) Z for all C E EIp. (5.14)

MIG:

Set

1: U14.

CEEIP

Since each are 1C has a fixed length it is clear that there exist a finite number of

pairwise disjoint arcs {11, ..., Im} of the unit circle such that,

N ow for each are I,- let S.- be a corresponding smooth sector such that

50

Let T, be the Riemann map from U onto 3,. Since S,- is a smooth sector there exists

15,- > 0 such that

l1m1nf NT’(Z) > 15,-. (5.15)
|z|—21",z€S,- logI—zl I

 

Set, S = U S,- and SC 2 U \ S. By construction we see

i=1

 

11m sup N‘”(Z,) = 0. (5.16)
|z|—+1,z€S¢ logl I—ZI

Thus there exist an 0 < r < 1 such that

1
N,p(z) g elogm (z 6 SC and r < |z| <1.) (5.17)

Now split the integral I”, into two integrals: one over Sf = S fl(U \ rU) and the

other over S to obtain,

LIUITI(C.I.f.)I2N.dA< / +/I1 2f.)I2NdA

We now consider each integral separately. In the estimate below of the integral over
Sf, we apply inequality (5.17) and then the Littlewood-Paley identity (2.2) for the

H 2 norm:

:1: I a: I 1
[55 |(C1lifn) IchpdA(Z) S C/U [(wan) I2 log md/fiz)

. (“0.21.1 — |2(C.7.fn)(0)|2)

 

|/\

€”(XIII-

51

Thus the integral over S? is a constant multiple of c, with the constant independent
of n.

We now turn our attention to estimating the integral over S. Using the fact that

S: U S, we see,

I. I1C2f.)I.12NdAz =2] I1)c2f.I2NdA1)15.18)

By Littlewood’s inequality (2.6) in Section 2.5 we see,

[5.- I(C,7,fn)’|2N,pdA(Z) S [31 |(C12fn)'(z)l2log I—:—IdA(z). (5.19)

By inequality (5.15) there exist 0 < r’ < 1 and a constant C such that

1
logm < CNT,(z) (z E S.- and r' < |z| <1)

for all 0 S i g m. Thus,

/ ((0211,) (z)|2log-I1—IdA() )<o/II( .‘;.)f. ’,,I2N dA(z) (5.21))

for all 0 S i S m. To the right side of equation (5.20) using the fact that T,(U) = S

and applying succesively the inequalities (5.19) and (5.20) we obtain,

/I1 1..) I2N dA <02: (.1) .’;.)f. I)'2N..dA1z.) 1521)

Applying to each term on the right side of (5.21) the change of variables formula (2.4)

and then the Littlewood-Paley identity (2.2) for the H 2 norm we obtain,

52

* ’ 5 l 1
[51(1) I(C1pfn) I2N¢dA(Z) S /(‘J |(CTij) (ZII210gmdA(z)

l/\
NIH

(IIC.C,2f.II2 — Io.,c,2f,10)I2).

Hence

<2
-2

Lu (0;,nt )I2Nw dA(z ;(|IC,,C;IntI2_ IC,,C,2f,,(0)I2).

Since dist(E,I,,r,(U))— — dist(E,I,I,-) > 6/2 where 2r,(U )flBU— — 1,, we see by Theo-
rem 5.6 that CIpC; is compact. Since CIIC; is compact its adjoint CTICJ, is compact.
Hence there exist an integer N such that
gifllC C21 II2—Ic 0'1 W) <. 1n>N)
2 Ti 1]) n T," 1p 11 ,

i=1

and this concludes the proof.

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BIBLIOGRAPHY

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