数理解析研究所講究録
1519 巻 2006 年 146-155

146

Hankel-type operators on the spacesof analytic functions
日本工業大学. 工学部

修–(Sh\^uichi Ohno)
Nippon Institute of Technology
大野Abstract. We will study Hankel-type operators on the spacesof analytic functions on the open unit disk. These operators
are a natural generalization of the classical Hankel operator
on the Hilbert Hardy space. They are related to tight uniform
algebras, the Dunford-Pettis property, and Bourgain algebras.

Introduction

1

Let $X$ be a Banach space and a closed subspace of $X$ . For an elementby:, we define the operator
such that$Y$$g$$g\mathrm{Y}\subset X$$S_{g}$$Yarrow X/\mathrm{Y}$$S_{\mathit{9}}f=gf+Y$

for all $f\in Y$ . The norm is considered as the quotient norm, that is,$||S_{g}f||=||gf+ \mathrm{Y}||=\inf\{||gf+h|| :h\in Y\}$for all $f\in Y$ . The quotient norm is the distance from $gf$ to $Y:d(gf, Y)=$$\inf\{||gf+h|| : h\in Y\}$ . This operator is called a Hankel-type operator
and is a natural generalization of the classical Hankel operator on theis said to be (weakly) compact if
Hilbert Hardy space. Recall thatmaps every bounded set into a relatively (weakly) compact one, and thatis said to be completely continuous if maps every weakly convergent
sequence into a norm convergent one. In general, every compact operator
is completely continuous. But the converse is not always true. We define
the following sets of symbols;$S_{g}$$S_{g}$$S_{g}$$S_{g}$

The author is partially supported by Grant-iIl-Aid for Scientific Research(No.17540169), Japan Society for the Promotion of Science.

147$Y_{c}=${$Y_{wC}=$$Y_{\mathrm{c}c}=$

is compact},$g:S_{g}$$g:S_{g}$is weakly compact},$g:S_{g}$is completely continuous}.

{{

to be compact, weakly compact and completelyfor
The conditi
continuous have been investigated in various function spaces. The probare weakly compact on a uniform algebra is relem of whether all
lated to a tight algebra [3] and the problem of complete continuity appears in the Dunford-Pettis property. The latter introduced a notion ofBourgain algebras which have been actively researched in analytic andharmonic function spaces on the open unit disk ([1], [2] and [10]). Recently, Dudziak, Gamelin, and Gorkin [5] studied Hankel-type operatorson analytic function spaces and Izuchi and the author [9] investiga,ted
Hankel-type operators on the space of bounded harmonic functions on
the unit disk. See [8] and [13] as surveys for convenience.
We here consider Hankel-type operators on the spaces of analytic functions on the open unit disk, explicltly, the disk algebra, Hardy andBergman spaces.
its boundary.
Let be the open unit disk in the complex plane andandbe the algebras of all continuous functions on
andLet
be the disk algebra of all continuous functions
respectively. Let
is the Banach algebra with theon that are analytic on D. Thensupremum norm.
$S_{g}$$o\mathrm{n}\mathrm{s}$$S_{g}$$\partial \mathrm{D}$$\mathrm{D}$$\partial \mathrm{D}$$C(\partial \mathrm{D})$$C(\overline{\mathrm{D}})$$A(\mathrm{D})$$\overline{\mathrm{D}}$$A(\mathrm{D})$$\overline{\mathrm{D}}$$||f||_{\infty}= \sup\{|f(z)|;z\in\overline{\mathrm{D}}\}$

be the Lebesgue spaces on
and
For $1\leq p\leq\infty$ , let
the classical Hardy
and respectively. For $1\leq p<\infty$ , we denote byspace that is the Banach space of all analytic function on for which$\partial \mathrm{D}$$L^{p}(\partial \mathrm{D})$$L^{p}(\mathrm{D})$$H^{p}$$\mathrm{D}$$\mathrm{D}$$f$,$||f||_{H^{p}}=(0 \mathrm{s}^{\backslash }\mathrm{u}\mathrm{p}\frac{1}{2\pi}\leq \mathrm{r}<1\int_{0}^{2\pi}|f(re^{i\theta})|^{p}d\theta)^{1/p}<\infty$

the Bergman space consisting of all analytic function
and denote by
on for which$If_{a}$$f$$\mathrm{D}$$||f||_{L_{a}^{p}}=( \int_{\mathrm{D}}|f(z)|^{p}dA(z))^{1/\mathrm{p}}<\infty$148
be the algebra ofwhere $dA$ is the normalized area measure on D. Let
bounded analytic functions on D. See [6], [7] and [15] for more informationon the Hardy and Bergman spaces.
as a closed subalgeIn the next section, we regard the disk algebraor
as a closed subalgebra ofandor
bra of:respectively. In section 3, we will consider the case of Hardy space:by, we define the linear operator
for:is a bounded$S_{g}f=gf+H^{p}$ for $f\in H^{p}.$ Tr.iviallybe the classical Hankel operator
linear operator. When $p=2$ , let$H_{g}f=gf-P(gf)$ , whereis the orthogonal
on ; for. It is well known thatis compact if
onto
projection from. In section 3, Theorem 3.1 says that thisand only iffor $1<p<\infty$ . When $p=1$ , Janson, Peetre andequivalence holds on
where
Semmes [11] studied the Hankel operator as form.
onto
is analytic polynomial and is the orthogonal projection of
from t.hc another
So we will give the attention to Hankel operators on
approach.
On the other hand, in the case of Bergman space, Leucking [12] char$1<p<\infty$ . So weacterized the compactness of Hankel operators on
will note them in section 4. In section 5 we add the result on the space ofbounded harmonic functions and in the last section we pose some open
questions.
$H^{\infty}$$A(\mathrm{D})$$H^{\infty}$$C(\overline{\mathrm{D}})$$C(\partial \mathrm{D})$$L^{\infty}(\partial \mathrm{D})$$L^{\infty}(\mathrm{D})$$H^{p}$$S_{g}$$g\in L^{\infty}(\partial \mathrm{D})$$H^{p}arrow L^{p}(\partial \mathrm{D})/H^{p}$$H^{p}arrow L^{p}(\partial \mathrm{D})/H^{p}$$S_{g}$$H_{g}$$H^{2}$$P$$g\in L^{\infty}(\partial \mathrm{D}),$$H^{2}$$L^{2}(\partial \mathrm{D})$$H_{g}$$g\in H^{\infty}+C(\partial \mathrm{D})$$H^{p}$$H_{b}f=\overline{P}(bf)$$L^{2}$$\overline{P}$$f$$\overline{H^{2}}$$H^{p}$$L_{a}^{p},$2

The disk algebra and(1) The disk algebra$H^{\infty}$$A(\mathrm{D})$as a closed subalgebra
At first we regard the disk algebra, we define the linear operator: $Aarrow$
of
. For
by
$S_{g}f=gf+A$ for $f\in A$ .
$A=A(\mathrm{D})|_{\partial \mathrm{D}}$$C(\partial \mathrm{D})$$g\in C(\partial \mathrm{D})$$S_{g}$$C(\partial \mathrm{D})/A$

Then we would characterize the sets.
subalgebra of$C(\partial \mathrm{D})$$A_{c},$$A_{wc}$and$A_{c\mathrm{c}}$. Each set is a closed

149
Theorem 2.1. When we regard, th en$A=A(\mathrm{D})|_{\partial \mathrm{D}}$as a closed su balgebra of$C(\partial \mathrm{D})$$A_{c}=A_{wc},=A_{cc}=C(\partial \mathrm{D})$.

.
and that. Then $|f_{n}(0)|\leq 1$ . Thus there exists a
Let $f_{n}\in A$ withsubsequence (which we do not relabel) of $\{f_{n}(0)\}$ such that $f_{n}(0)arrow c$ for,some constant . Then, since

Proof. It is trivial that.

$A\subset A_{c}$$A_{c}\subset A_{\tau\iota’ c}$$\subseteq A_{cc}\subset C(\partial \mathrm{D})$$||f_{n}||_{\infty}\leq 1$$\overline{z}(f_{n}-f_{n}(0))\in A$$c$$||\overline{z}f_{n}-\overline{z}c+A||_{\infty}$$\leq||\overline{z}f_{n}-\overline{z}c-\overline{z}(f_{n}-f_{n}(0))||_{\infty}$$=||\overline{z}(c-f_{n}(0))||_{\infty}$$=|c-f_{n}(0)|arrow 0$ .

So

$S_{\overline{z}}$is compact and$\overline{z}\in A_{c}$. Because$A_{c}$

is a closed subalgebra of$A_{\mathrm{c}}=A_{wc}=A_{cc}=C(\partial \mathrm{D})$$C(\partial \mathrm{D})$,,by the Stone-Weierstrass theorem.

$\square$

We here estimate the norm and the essential norm of . Recall that
to $X/Y$ isfromthe essential norm of a bounded linear operator
defined as$S_{g}$$Y$$T$$||T||_{e}= \inf${ $||T+K||$ :$K$

is compact operator from $Y$ to $X/Y$ }.

Using the basic duality relation ([7: Chapter IV]), we has the following.
Theorem 2.2. For$g\in C(\partial \mathrm{D})$, thenand $||S_{g}||_{e}=0$ .

$||S_{\mathit{9}}||=d(g, A(\mathrm{D})|_{\partial \mathrm{D}})$

as a closed subalgebra ofSecondly we regardby:, we define the linear operator$A=\mathrm{A}(\mathrm{D})$$g\in C(\overline{\mathrm{D}})$$S_{\mathit{9}}$$S_{\mathit{9}}f=gf+A$

for$A_{wc}$$f\in A$

$A_{cc}$Theorem 2.3. When we regardthen$A(\mathrm{D})$. For$Aarrow C(\overline{\mathrm{D}})/A$.

Then each set
andis a closed subalgebra ofequivalence was proved by Cole and Gamelin [3].
$A_{c},$$C(\overline{\mathrm{D}})$$C(\overline{\mathrm{D}})$and their

as a closed su balgebra of$A_{c}=A_{wc}=A_{cc}=C(\overline{\mathrm{D}})$$C(\overline{\mathrm{D}})$,.

In this case we also have the following using the basic duality relation.

150
Theorem 2.4. For(2)$g\in C(\overline{\mathrm{D}})$, th en$||S_{\mathit{9}}||=d(g, A(\mathrm{D}))$and$||S_{\mathit{9}}||_{e}=0$.

$H^{\infty}$

as a closed subalgebra of
At first we regard:, we define the linear operator$H^{\infty}$$H^{\infty}$$S_{g}$$L^{\infty}(\partial \mathrm{D})$for$S_{g}f=gf+H^{\infty}$—$L^{\infty}(\partial \mathrm{D})$.

$L^{\infty}(\partial \mathrm{D})/H^{\infty}$$f\in H^{\infty}$

Forby$\mathit{9}\in$.

has the Dunford-Pettis property, Cima,
Then using the fact that
Janson and Yale [1] and Gorkin [8] showed the following.
$H^{\infty}$

Theorem 2.5. When we regardthen

as a closed su balgebra of$H^{\infty}$$H_{c}^{\infty}=H_{wc}^{\infty}=H_{c\mathrm{c}}^{\infty}=H^{\infty}+C(\partial \mathrm{D})$$L^{\infty}(\partial \mathrm{D})$,.

The estimation of norms is the following.

Theorem 2.6. For.

$g\in L^{\infty}(\partial \mathrm{D})_{f}$then $||S_{g}||=d(g, H^{\infty})$ and$||S_{\mathit{9}}||_{e}\leq$$d(g, H^{\infty}+C(\partial \mathrm{D}))$

as a closed subalgebra ofSecondly we regard:, we define the linear operator$H^{\infty}$$L^{\infty}(\mathrm{D})$$H^{\infty}arrow L^{\infty}(\mathrm{D})/H^{\infty}$$S_{\mathit{9}}$$S_{g}f=gf+H^{\infty}$$L^{\infty}(\mathrm{D})$for. For $g\in$by$f\in H^{\infty}$

Then Cima, Stroethoff and Yale [2] obtained the following result.
Theorem 2.7. When we regard
tben$H^{\infty}$

as a closed su balgebra of$L^{\infty}(\mathrm{D})$,$H_{c}^{\infty}=H_{wC}^{\infty}=H_{cc}^{\infty}=H^{\infty}+C(\overline{\mathrm{D}})+V$

where $V=$ {$g\in L^{\infty}(\mathrm{D}):||g\chi_{\mathrm{D}\backslash r\mathrm{D}}||arrow 0$

as$rarrow 1^{-}$}.

Fbrthermore the estimation of norms is the following.
Theorem 2.8. For$g\in L^{\infty}(\mathrm{D})$$d(g, H^{\infty}+C(\overline{\mathrm{D}})+V)$.

, then$||S_{\mathit{9}}||=d(g, H^{\infty})$and$||S_{g}||_{e}\leq$151

3The case of Hardy spaces$H^{p}$

for $1<p<$$\infty$

We here consider the case of Hardy spaces. Before starting our discus.
sion, we recall results concerning the topology of Hardy spaces$H^{p}$

Fact 1. ([4: Chap.20, Proposition 3.15]) If $1<p<\infty,$ $f\in H^{p}$ , and, then the following are equivalent.
is a sequence inconverges weakly to
.
(a)
(b)
is bounded and $f_{n}\in H^{p}$ converges to uniformly on every
compact subset of D.
(c)converges to $f(z)$ for all $z\in$ D.
is bounded andconverges to $f^{k}(0)$ for all $k\geq 0$ .
(d)is bounded and$f_{n}$$H^{\mathrm{p}}$$\{f_{n}\}$$f\in ff^{\mathrm{p}}$$f$$\{f_{n}\}$$\{f_{n}\}$$f_{n}(z)$$\{f_{n}\}$$f_{n}^{k}(0)$Fact 2. ([4: Chap.20, Proposition 3.16]) Put $S=\{f\in H^{1} : ||f||_{H^{1}}=1\}$ .
Then is weak*-compact and metrizable, but not weak compact.
For $f_{n}\in S$ , the following are equivalent:.
(i)converges to in the weak*-topology in(ii) $f_{n}(z)$ converges to $f(z)$ for all $z\in$ D.
converges to uniformly on every compact subset of D.
(iii)
$S$

$H^{1}$$f$$f_{n}$$f$$f_{n}$For $1\leq p<\infty$ andby$g\in L^{\infty}(\partial \mathrm{D}))$

we define the linear operator$S_{g}$:$H^{p}arrow L^{p}(\mathrm{D})/H^{p}$$S_{g}f=gf+H^{p}$for$f\in H^{p}$.

, the following are equivalent:
Fact 3. For $1<p<\infty$ and(i):is compact (completely continuous).
(ii) If
and converges to uniformly on every
is bounded in.
compact subset of , then$g\in L^{\infty}(\partial \mathrm{D})$$S_{g}$$H^{p}arrow L^{p}(\partial \mathrm{D})/H^{p}$$0$$H^{\mathrm{p}}$$\{f_{n}\}$$||S_{g}f_{n}||arrow 0$$\mathrm{D}$For $1<p<\infty,$
is reflexive. So all completely continuous operator
is always weaklyonis compact and every bounded operator on.
compact. Thusanddefined, let
For
be the classical Hankel operator on
by $H_{g}f=gf-P(gf)$ , where is the orthogonal projection from:is compact if. Hartman’s theorem says thatonto
is bounded from. Then for $1<p<\infty,$
and only if$H^{p}$$H^{p}$$H^{p}$$H_{c}^{p}=H_{cc}^{p}$$g\in L^{\infty}(\partial \mathrm{D})$$H_{wc}^{p}=L^{\infty}(\partial \mathrm{D})$$H^{2}$$H_{g}$$P$$H^{2}$$L^{2}(\partial \mathrm{D})$$H_{g}$$g\in H^{\infty}+C(\partial \mathrm{D})$$H^{2}arrow L^{2}(\partial \mathrm{D})$

$P$

152
and we can easily see the equivalence of compacteness ofand . But the next result will give the characterization of Hankel
from the another approach.
operators on$L^{p}(\partial \mathrm{D})$onto$H^{p}$$S_{g}$$H_{g}$$H^{p}$

Theorem 3.1. For $1<p<\infty$ , the following hold:and$H_{c}^{p}=H_{cc}^{p}=H^{\infty}+C(\partial \mathrm{D})$Proof. First, we note that

$H_{w\mathrm{c}}^{\mathrm{p}}=L^{\infty}(\partial \mathrm{D})$$H^{\infty}\subset H_{c}^{p}=H_{\mathrm{c}c}^{p}\subset L^{\infty}(\partial \mathrm{D})$.

. Then $B:=H_{c}^{p}=$is a closed algebra and so a Douglas algebra.
. Thus there exists an interpolatSuppose that
. That is,withis compact
ing Blaschke productwhere $b_{n}(z)=$(completely continuous). Write. Put. Then. By Fact 1, $f_{k}(z)arrow 0$ weakly in.
as
and $f_{k}(z)arrow 0$ for
On the other hand, we have$H_{c}^{\mathrm{p}_{\mathrm{C}}}$$H^{\infty}+C(\partial \mathrm{D})\subset\wedge B$$\overline{\psi}\in B$$\psi\in H^{\infty}$$S_{\overline{\psi}}$$\psi(z)=e^{i\alpha}\prod_{n=1}^{\infty}b_{n}(z)$$(z-’\sim’ n)/(1-\overline{z_{n}}z)$$f_{k}\in H^{p},$ $||f_{k}||_{H^{p}}=1$$f_{k}(z)= \prod_{n=k}^{\infty}b_{n}(z)$$z\in \mathrm{D}$$karrow\infty$$H^{p}$$||S_{\overline{\psi}}f_{k}||=||\overline{\psi}f_{k,}+H^{p}||$$=||\overline{e^{i\alpha}b_{1}b_{2}\cdots b_{k-1}}+H^{p}||$$=f\in \mathrm{i}1\mathrm{u}\mathrm{f}_{\mathrm{p}}||1+e^{i\alpha}b_{1}b_{2}\cdots b_{k-1}f||_{H^{\mathrm{p}}}$$\geq\inf_{f\in H^{p}}|1+e^{i\alpha}(b_{1}b_{2}\cdots b_{k-1}f)(z)|(1-|z|^{2})^{1/p}$.
for
Put $z=z_{1}$ , a zero of,$z\in \mathrm{D}$$b_{1}$. So$||S_{\overline{\psi}}f_{k}||\geq(1-|z_{1}|^{2})^{1/q}>0$As$S_{\overline{\psi}}$.

is completely continuous,$||S_{\overline{\psi}}f_{k}||arrow 0$This contradicts. So$B=H^{\infty}+C(\partial \mathrm{D})$.

.

$\square$Furthermore the estimation of norms is the following.
Theorem 3.2. For $1\leq p<\infty$ andand for $1<p<\infty,$$g\in L^{\infty}(\mathrm{D})$$||S_{g}||_{e}\leq d(g, H^{\infty}+C(\partial \mathrm{D}))$.

, then$||S_{\mathit{9}}||=d(g, H^{\infty})$153

4The case of Bergman spaces$L_{a}^{p}$for $1<$$p<\infty$, let
be the classical Hankel operator defined by$H_{\mathit{9}}f=gf-P(gf)$ , where $P$ is the Bergman projection fromonto .

For$g\in L^{\infty}(\mathrm{D})$$H_{g}$$L^{p}$$L_{a}^{p}$and .
Then we can easily see the equivalence of compacteness of
On the other hand, Leucking [12] characterized the compactness of Hankel$1<p<\infty$ . And so we have the following.
operators on$H_{g}$$S_{g}$$L_{a}^{p},$Theorem 4.1. For $1<p<\infty$ , thenadmits a decomposition $g=g_{1}+g_{2}$ so that$g\in(L_{a}^{p})_{c}=(L_{a}^{p})_{cc}$if and only if$g$$\lim_{|z|arrow 1}\frac{1}{|D(z)|}\int_{D(z)}|g_{1}|^{2}dA=0$

and$g_{2}\in C^{1}(\mathrm{D})$,$\lim_{|z|arrow 1}(1-|z|)\overline{\partial}g_{2}(z)=0$

where $D(z)$ is the Bergman disk with cent er .
.
Moreover it holds that$z$$(L_{a}^{p})_{wc}=L^{\infty}(\mathrm{D})$5 The space of bounded harmonic functions
We here consider Hankel-type operators on the space of bounded harmonic functions. Let
be the set of all bounded harmonic. We
functions on D. It follows thatis a closed subspace ofis charcan define
as before. The Bourgain algebra
andacterized by Izuchi, Stroethoff and Yale [10].
For a function, we denote by the Poisson integral ofon , that is,$h^{\infty}:=h^{\infty}(\mathrm{D})$$h^{\infty}$$h_{\mathrm{c}}^{\infty},$$h_{wc}^{\infty}$$L^{\infty}(\mathrm{D})$$h_{cc}^{\infty}$$h_{c\mathrm{c}}^{\infty}$$f$$\hat{f}$$f\in L^{\infty}(\partial \mathrm{D})$$\mathrm{D}$$\hat{f}(z)=\int_{0}^{2\pi}f(e^{i\theta})P_{z}(e^{i\theta})d\theta/2\pi$,. For any nonempty. Thenwhere is the Poisson kernel of, we write $\hat{B}=\{\hat{f} : f\in B\}$ . It is known that
subset , ofon , so thatin
has a boundary function
and
on
. Let
be the space of boundary functions of$B$$\hat{f}\in h^{\infty}$$z\in \mathrm{D}$$P_{z}$$f$$L^{\infty}(\partial \mathrm{D})$$h^{\infty}$$h^{\infty}=L^{\infty}\overline{(\partial \mathrm{D}})$$f^{*}$$H^{\infty}(\partial \mathrm{D})$$\partial \mathrm{D}$$f^{*}\wedge=f$$\mathrm{D}$154bounded analytic functions on D. The algebra $QC$ of bounded quasiis given byfunctions on
continu$\partial \mathrm{D}$$o\mathrm{u}\mathrm{s}$.

$QC=(H^{\infty}(\partial \mathrm{D})+C(\partial \mathrm{D}))\cap\overline{(H^{\infty}(\partial \mathrm{D})+C(\partial \mathrm{D}))}$

Refer to [7] and [14] for more information.
was given as Corollary 3 in [10], where
The equality
is the same set as in Theorem 2.7. Then Izuchi and the author [9] show
the following result.
$V$$h_{cc}^{\infty}=\overline{QC}+V$

Theorem 5.1.

.

$h_{c}^{\infty}=h_{\mathrm{c}c}^{\infty}.=\overline{QC}+V$

Problems

6

Problem 6.1. Estimate the essenti norms of Hankel-type operators incases of Hardy and Bergman spaces.
$al$Problem 6.2. How about the case $p=1.$? That is, what are$H_{c}^{1},$$H_{wc}^{1}.$,.

$H_{cc}^{17}$Problem 6.3. Let
D. Then characteriz$h^{\infty}$

be the space of bounded harmonic functions on$eh_{wc}^{\infty}$.

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[15] K. Zhu, Operator Theory in Function Spaces, Marcel Dekker, New
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Nippon Institute of Technology, Miyashiro, Minami-Saitama 345-8501,
Japan-mail address: ohno@nit.ac.jp$\mathrm{B}$