数理解析研究所講究録 1277 巻 2002 年 106-119

106
COMPACT DIFFERENCES OF TWO COMPOSITIONOPERATORS

日本工業大学 工学部

大野

・

修一

(Shuichi Ohno)

Nippon Institute of Technology

1. INTRODUCTION
Throughout this article, we let D be the open unit disk andboundary. Let dm denote the normalized Lebesgue measure on
for $0<p\leq\infty$ . Let

denote the classical Hardy space by$H^{p}$of all analytic self-maps of D. Every$\varphi\in S(\mathrm{D})$alinear composition operator$C_{\varphi}$. Thus$C_{\varphi}f=f$$C_{\varphi}$o$S(\mathrm{D})$$\partial \mathrm{D}$$\partial \mathrm{D}$

its

. We

be the set

induces through composition

is defined by$\varphi$

for analytic function f on D. By the Littlewood’s subordination theorem,$C_{\varphi}$

is abounded operator on$H^{2}$.

Many authors have investigated some properties of composition operators and tried to characterize such properties of the operators

functional analytic properties of its symbol$\varphi$$C_{\varphi}$using. Here we will give arepor107
on the problem when the difference of two composition operators would becompact on$H^{2}$. For ageneral information on composition operators, see[4],[15] and [17: Chaper 10].

2. THE DEVELOPMENT
The work originates from the following result of E. Berkson([l]).
[E. Berkson(1981)] Let$\{|\varphi|=1\}$$\varphi\in S(\mathrm{D})$such that $m(E(\varphi))>0$ , where $E(\varphi)=$. If $||C_{\varphi}-C_{\psi}||^{2}<m(E(\varphi))/2$ for$\psi$$\in S(\mathrm{D})$, then$\varphi=\psi$.

This result makes atopological statement about the space $C(H^{2})$ of com-position operators on$H^{2}$, endowed with the operator norm metric. Indeed

this says that the identity operator is isolated in $C(H^{2})$ .

A. Siskakis (1986) asked if every non-compact composition operator had
to be isolated in the space $C(H^{2})$ . Then it was begun to explore the groundthat lies between the compactness and the isolation in $C(H^{2})$ , and the ques-tion above had anegative answer later ([16]).
$\mathrm{B}.\mathrm{D}$. MacCluer ([9]) gave asufficient condition on

containing the composition operator

An analytic map
point$\zeta\in\partial \mathrm{D}$$\varphi\in S(\mathrm{D})$if there exists$C_{\varphi}$$\varphi$

for the component

to be the singleton$\{C_{\varphi}\}$.

is said to have an angular derivative at a$w\in\partial \mathrm{D}$so that the non-tangential limit$\lim\underline{\varphi(z)-w}$$zarrow\zeta$$z-\zeta$

108[B.D. MacCluer (1989)] If
positive measure, then$C_{\varphi}$$\varphi$

has afinite angular derivative on aset ofis isolated in $C(H^{2})$ .

J.H. Shapiro and C. Sundberg ([16]) explored these territory and gave anumber of conjectures:

1. Characterize the components of $C(H^{2})$ .

2. Which composition operators are isolated in$C(H^{2})^{q}$

3. Which composition differences are compact on$H^{2q}$They supposed that two composition operators may belong to the same
component of $C(H^{2})$ if and only if they differ by acompact. They offeredsome sort of joint Nevanlinna counting functions figuring into the problem.
They gave the following result to the isolation problem.
[J.H. Shapiro and C. Sundberg (1990)] If$\varphi\in S(\mathrm{D})$satisfies$\int\log(1-|\varphi|)dm>-\infty$then$C_{\varphi}$

is not isolated in $C(H^{2})$ .

It is well known that the condition above characterizes the non- extreme
point of the unit ball of $H^{\infty}([5])$ . So by Berkson’s result and this we canreduce that if$\varphi$

is an exposed point of the unit ball of$H^{\infty}$, then$C_{\varphi}$

is

109
isolated in $C(H^{2})$ and that ifof the unit ball of$H^{\infty}$

is isolated in $C(H^{2})$ ,$C_{\varphi}$

is an extreme point$\varphi$.

Moreover this hinges afollowing sufficient condition for the difference tobe compact.
[J.H. Shapiro and C. Sundberg (1990)] If, for$\varphi$,$\psi$$\int\frac{|\varphi-\psi|}{(\min\{1-|\varphi|,1-|\psi|\})^{3}}dm<\infty$then$C_{\varphi}-C_{\psi}$

is compact on$H^{2}$$\in S(\mathrm{D})$

,

,

.

H. Hunziker, H. Jarchow and V. Mascioni([7]) defined the following met$\mathrm{r}\mathrm{i}\mathrm{c}$in $C(H^{2})$ and called the topology induced by this the Hilbert-Schmidt

topology: for$\varphi$,$\psi\in S(\mathrm{D})$,$d( \varphi, \psi)=(\frac{1}{2\pi}\int_{0}^{2\pi}|\frac{\varphi-\psi}{1-\overline{\varphi}\psi}|^{2}\frac{1-|\varphi|^{2}|\psi|^{2}}{(1-|\varphi|^{2})(1-|\psi|^{2})}d\theta)^{1/2}$

And they gave the result.
[H. Hunziker, H. Jarchow and V. Mascioni(1990)] For

following are equivalent:(i)$\varphi$

(ii)
(iii)

$\varphi$

is an extreme point of the unit ball ofis isolated in$C_{\varphi}$$(S(\mathrm{D}), d)$

is isolated in$S(\mathrm{D})$.

;$H^{\infty}$;$\varphi\in S(\mathrm{D})$

, the

110
3. NEW RESULTS
Recently some authors have attacked these problems using new tools.

In this section we summarize them.

In 1997, J.A. Cima and A.L. Matheson ([3]) characterized the essential norm ||$||_{e}$of composition operators using the notion of Aleksandrov

measures.
For$\varphi\in S(\mathrm{D})$and A$\in \mathrm{D}$, there exists apositive measure$\mu_{\lambda}$on$\partial \mathrm{D}$

such

that

${\rm Re} \frac{\lambda+\varphi(z)}{\lambda-\varphi(z)}=\frac{1-|\varphi(z)|^{2}}{|\lambda-\varphi(z)|^{2}}$$= \int P(\zeta, z)d\mu_{\lambda}(\zeta)$,

where P(., z) is the Poisson kernel for z,$P( \zeta, z)=\frac{1-|z|^{2}}{|\zeta-z|^{2}}$Then$\mu_{\lambda}$.

is called the Aleksandrov measure with the function

the absolutely continuous part and the sigular part ofrespectively.
[J.A. Cima and A.L. Matheson(1997)$||C_{\varphi}||_{e}^{2}= \sup${$||\mu_{\lambda}^{s}||$:A$\in\partial \mathrm{D}$}.

$\mu_{\lambda}$by$\varphi$. Denote$\mu_{\lambda}^{a,c}$and$\mu_{\lambda}^{s}$

111
This result has the immediate corollary:
if for all$\lambda\in\partial \mathrm{D}\mu_{\lambda}$$C_{\varphi}$

is compact on$H^{2}$

if and onlyis absolutely continous with respect to the Lebesgue

measure $dm$ .
Then$\mathrm{J}.\mathrm{E}$. Shapiro ([12]) considered the compact difference using thisnotion.
[J.E. ShapirO(1998)] If

for all A$\in\partial \mathrm{D}$$C_{\varphi}-C_{\psi}$

is compact on$H^{2}$for ,$\varphi$$\psi\in S(\mathrm{D})$,$\mu_{\lambda}^{s}=\nu_{\lambda}^{s}$.

He conjectured whether its converse would be true.
But it does not seem to be easy to calculate Aleksandrov measure withrespect to any self-map of D.

[Example 1] Let $\varphi(z)=sz+(1-s)z$ for $0<s<1$ . Let$\mu_{\lambda}$

be theAleksandrov measure with the function .
$\varphi$

Then we have$||\mu_{\lambda}^{s}||=||\mu_{\lambda}||-||\mu_{\lambda}^{a,c}||$$= \frac{1-|\varphi(0)|^{2}}{|\lambda-\varphi(0)|^{2}}-\int\frac{1-|\varphi(\zeta)|^{2}}{|\lambda-\varphi(\zeta)|^{2}}dm(\zeta)$.

Putting $\lambda=1$ , we have the first term of the right side is $(2-s)/s$ andthe second term is $(1-s)/s$ . So $||\mu_{1}^{s}||=1/s>0$ . Consequently
compact on$H^{2}$.

$C_{\varphi}$

is not112
These measures have played an interesting role in the study of the deBranges-Rovnyak space. J.E. Shapiro has provided the study of relative

angular derivatives ([13], [14]).

T.E Goeber, Jr. ([6]) connected this problem with the compactness ofcomposition operators between different Hardy spaces.

Let $0<q<p<\infty$ . Thenfor$\varphi\in S(\mathrm{D})$is always bounded ffom

$C_{\varphi}$$H^{p}$

to$H^{q}$

andare compact from

$C_{\psi}$$H^{q}$.

[T.E Goeber, Jr.(2001)] For $0<q<p<\infty$ ,$C_{\varphi}$to. He characterized the essential norm of differences of two

composition operators ffom

if$H^{p}$$H^{p}$

to$H^{q}$$||C_{\varphi}-C_{\psi}||_{e}=0$

if and only.

And he offered the folowing conjecture :Let $0<q<p<\infty$ . Is ittrue that$C_{\varphi}$,$C_{\psi}$

operators ffom

are in the same component of the space of composition$H^{p}$

to$H^{q}$

if and only if$C_{\varphi}$,$C_{\psi}$

are compact ffom$H^{p}$

to$H^{q?}$

Indeed this result inspires us to consider one question:[Question] What is the space X of analytic functions on D satisfying that
$C_{\varphi}-C_{\psi}$:X$arrow H^{2}$is compact if and only if$C_{\varphi}-C_{\psi}$: $H^{2}arrow H^{2}$ is compact?

When B.D. MacCluer, S. Ohno and R. Zhao ([11]) reduce the problem

of compact difference to the$H^{\infty}$

case, they obtain the result:$C_{\varphi}-C_{\psi}$:

113$H^{\infty}arrow H^{\infty}$$B$

is compact if and only if$C_{\varphi}-C_{\psi}$:$5arrow H^{\infty}$

is compact, whereis the Bloch space.

So we can suppose the Bloch space as acandidate of the answer to the
problem above. But we can find out the interesting result due to E.G. Kwon([8]):[E.G. Kwon (1996)] For

if$\varphi$$\varphi\in S(\mathrm{D})$,$C_{\varphi}$:B$arrow H^{2}$

is not an extreme point of the unit ball of$H^{\infty}$$\int\log(1-|\varphi|)dm>-\infty$

is compact if and only

, that is,.

We here see again the condition of the non-extreme point of the unit ballof$H^{\infty}$, which appears in the problem of the hypercyclicity of composition

operators ([2]). This condition seems to be interesting and mysterious.

We have the following equivalence.
[Proposition] For(i)$C_{\varphi}-C_{\psi}$

(ii)

:

$C_{\varphi}-C_{\psi}$$\varphi$,$\psi\in S(\mathrm{D})$$B$ $arrow H^{2}$:$B$ $arrow H^{2}$, the following are equivalent:is bounded;

is compact;(iii)$C_{\varphi}-C_{\psi}$:$B_{o}arrow H^{2}$

is bounded;$(\mathrm{i})$$C_{\varphi}-C_{\psi}$:$B_{o}arrow H^{2}$is compact,where$B_{o}$

is the little Bloch space.

114
About the compact difference, we can find out two examples in [4]: Example 9.1 at p.336 says that for $\varphi(z)=(z+1)/2$ and $\psi(z)=\varphi(z)+t(z-1)^{3}$ ,$C_{\varphi}-C_{\psi}$

is compact on

On the other hand, Exercises 9.3.3 at p.344$H^{2}$

gives that for $\varphi(z)=(z+1)/2$ and $\psi(z)=\varphi(z)+t(z-1)^{2}$ ,not compact on$H^{2}$$C_{\varphi}-C_{\psi}$is

What exists between these two examples? We havecalculated but not completed.
Recently it is reported by B.D. MacCluer ([10]) that J. Moorhouse an-swers this as follows.
[Example 2] Let $\varphi(z)=sz+1-s$ and $\psi(z)=\varphi(z)+t(z-1)^{b}$ for fixed

real numbers s and t such that $0<s<1$ and$\psi(\mathrm{D})\subset \mathrm{D}$. Notice that |t| isso small. For apositive number 6,(i) In the case $0<b\leq 2$ ,$C_{\varphi}-C_{\psi}$(ii) In the case $2<b<5/2$ ,(iii) In the case $5/2<b$ ,is not Hilbert-Schmidt on$C_{\varphi}-C_{\psi}$$C_{\varphi}-C_{\psi}$

is compact on$H^{2}$

is Hilbert-Schmidt on

In the case of the Bergman space$L_{a}^{2}=L_{a}^{2}$$H^{2}$$H^{2}$(D, dA) where dA is thenormalized Area measure on D, we have the following incomplete result.
[Example 3] Under the same assumption as Example 2,(i) If $0<b\leq 2$ ,(ii) If $3<b$ ,$C_{\varphi}-C_{\psi}$$C_{\varphi}-C_{\psi}$

is not compact on

is compact on$L_{a}^{2}$.

$L_{a}^{2}$.

115
We will add the outline of the proof: (i) At first suppose $0<b<2$ . For

any A
and

$\in \mathrm{D}$$k_{\lambda}$, let$k_{\lambda}(z)=(1-|\lambda|^{2})/(1-\overline{\lambda}z)^{2}$

converges to 0weakly in$L_{a}^{2}$as. And then$|\lambda|arrow 1$$k_{\lambda}\in L_{a}^{2}$,$||k_{\lambda}||=1$. Then$(*)||(C_{\varphi}-C_{\psi})^{*}k_{\lambda}||^{2}$$=( \frac{1-|\lambda|^{2}}{1-|\varphi(\lambda)|^{2}})^{2}+(\frac{1-|\lambda|^{2}}{1-|\psi(\lambda)|^{2}})^{2}-2{\rm Re}(\frac{1-|\lambda|^{2}}{1-\overline{\varphi(\lambda)}\psi(\lambda)})^{2}$$\geq(\frac{1-|\lambda|^{2}}{1-|\varphi(\lambda)|^{2}})^{2}-2|\frac{1-|\lambda|^{2}}{1-\overline{\varphi(\lambda)}\psi(\lambda)}|^{2}$

We also consider for asequence$|1-\lambda_{n}|^{2}=1-|\lambda_{n}|^{2}$$\{\lambda_{n}\}\mathrm{o}\mathrm{f}$points approaching 1along the circle. Then we have.

$||(C_{\varphi}-C_{\psi})^{*}k_{\lambda_{n}}||^{2} \geq\frac{1}{(2-s)^{2}s^{2}}-\frac{2(1-|\lambda_{n}|^{2})^{2-b}}{|(2-s)s(1-|\lambda_{n}|^{2})^{1-b/2}-|t\varphi(\lambda_{n})||^{2}}$Consequently$\lim\{||(C_{\varphi}-C_{\psi})^{*}k_{\lambda_{n}}||_{2}^{2} :that is,$C_{\varphi}-C_{\psi}$|\lambda_{n}|arrow 1, |1-\lambda_{n}|^{2}=1-|\lambda_{n}|^{2}\}\geq\frac{1}{(2-s)^{2}s^{2}}$is not compact on$L_{a}^{2}$,.

Secondly suppose $b=2$ . For asequence of points approaching 1along

the circle $|1-\lambda|^{2}=1-|\lambda|^{2}$ , we can calculate the right side of the equation(’) and show that

$C_{\varphi}-C_{\psi}$

is not compact on$L_{a}^{2}$.

116(ii) For afunction f$\in L_{a}^{2}$, we have$(C_{\varphi}-C_{\psi})f(z)$$= \int f(w)\mathrm{t}\frac{1}{(1-\varphi(z)\overline{w})^{2}}-\frac{1}{(1-\psi(z)\overline{w})^{2}}\}$dA(w)$= \int f(w)(\frac{1}{1-\varphi(z)\overline{w}}-\frac{1}{1-\psi(z)\overline{w}})$$\cross(\frac{1}{1-\varphi(z)\overline{w}}+\frac{1}{1-\psi(z)\overline{w}})$

dA(w)

So

$|(C_{\varphi}-C_{\psi})f(z)|^{2}$$\leq\int|f(w)|^{2}|\frac{1}{1-\varphi(z)\overline{w}}-\frac{1}{1-\psi(z)\overline{w}}|^{2}$$\cross\int|\frac{1}{1-\varphi(z)\overline{w}}+\frac{1}{1-\psi(z)\overline{w}}|^{2}$

dA(w)

dA(w)

$\leq\int|f(w)|^{2}|\frac{\varphi(z)-\psi(z)}{(1-\varphi(z)\overline{w})(1-\psi(z)\overline{w})}|^{2}$dA(w)$\mathrm{x}2\{\int|\frac{1}{1-\varphi(z)\overline{w}}|^{2}dA(w)+\int|\frac{1}{1-\psi(z)\overline{w}}|^{2}dA(w)\}$$\leq C\int|f(w)|^{2}dA(w)|t||z-1|^{2(b-4)}dA(w)$$\cross(\log\frac{1}{1-|\varphi(z)|^{2}}+\log\frac{1}{1-|\varphi(z)|^{2}})$where C is aconstant.
Using the facts that $\log 1/(1-|z|^{2})\in L^{p}$ for $0<p$ and $1/(z-1)\in L_{a}^{p}$for $0<p<2$ , we can show

$C_{\varphi}-C_{\psi}$

is compact on$L_{a}^{2}$

for $3<b$ .

117
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5293-5316.
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119
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