数理解析研究所講究録
第 1596 巻 2008 年 18-24$p$

18-Hyponormality of weighted composition operators
宗雄 (Muneo Ch\={o})

神奈川大学

長

神奈川大学

Kanagawa University
山崎丈明 (Takeaki Yamazaki)
Kanagawa University

ABSTRACT
Characterizations of the classes of p-hyponormal,-hyponormal and weakhyponormal weighted composition operators are introduced. It is shown thatsome classes of weak hyponormal weighted composition operators can not be
separated. It is an extension of the result by C. Burnap, etc.
$\infty$

This report is based on the following paper:[CY] M. Cho and T. Yamazaki, Characterizations of p-hyponormd and weakhyponormal weighted composition operators, preprint.

1. INTRODUCTION
be a complex Hilbert space, andbe the algebra of all bounded linear.
operators on . For a bounded linear operator, many authors have studied properties
of weak normality of operators, especially, the class of hyponormal operators defined
as follows:Let$\mathcal{H}$$B(\mathcal{H})$$\mathcal{H}$Definition 1. Let. Then the following operator classes are defined:(i)is $hyponorma1\Leftrightarrow T^{*}T\geq TT^{*}$ ,(ii) for $p>0,$ is $p- hyponorma1\Leftrightarrow(T^{*}T)^{P}\geq(TT^{*})^{p}$ .
Especially, if is p-hyponormal for all $p>0$ , we call-hyponormal ([10]).
$T\in B(\mathcal{H})$

$T$

$T$

$T\infty$$T$

It is well known that typical examples of these operators are expressed by polynomials of weighted shift operators on . So many authors have studied properties ofweighted shift operators. Examples are obtained as follows:$l^{2}$

Example 1.1. Let$U$be a weighted unilateral shift on$U=(^{1}0002002$$00$.
Then$0.$$\cdot..$$l^{2}$).

as follows:19(i)
is -hyponormal, not quasinormal (i.e., $T^{*}TT\neq TT^{*}T$ ),(ii) $2U+U^{*}$ is hyponormal, not -hyponormal,(iii) $(2U+U^{*})^{2}$ is -hyponormal, not hyponormal.
$U$$\infty$$\infty$$\frac{1}{2}$Moreover, classes of weak hyponormal operators have been also dePned, especially,
the following operator classes are studied by many authors, strenuously.

Definition 2. Let$T\in B(\mathcal{H})$. Then the following operator classes are defined:(i) $T$ is $quasihyponorma1\Leftrightarrow T^{*}T^{*}TT\geq T^{*}TT^{*}T$ ,(ii) for $p>0,$ $T$ is $pquasihyponorma1\Leftrightarrow T^{*}(T^{*}T)^{p}T\geq T^{*}(TT^{*})^{p}T$,(iii) $T$ belongs to class $A\Leftrightarrow|T^{2}|\geq|T|^{2}([4,5])$ ,(iv) for $t>0,$ $T$ belongs to class $A(s,t)$.
$s,$$\Leftrightarrow(|T^{*}|^{t}|T|^{2s}|T^{*}|^{t})^{\frac{t}{+}}\geq|T^{*}|^{2t}([3])$We remark that the class A coincides with the class $A(1,1)$ . These operator classes

are defined by operator inequalities. As parallel classes of them, the classes of paranormal and $absolute-(s, t)$ -paranormal operators are known.
Definition 3. Let. Then the following operator classes are defined:(i)([4]),
is paranormal
for all(ii) for $t>0,$ is $absolute-(s,t)$ -paranormal““([12]).
for all$T\in B(\mathcal{H})$$T$$\Leftrightarrow\Vert T^{2}x\Vert\Vert x\Vert\geq\Vert Tx\Vert^{2}$$s,$$x\in \mathcal{H}$$T$$\Leftrightarrow\Vert|T|^{s}|T$$|^{t}x\Vert^{t}\Vert x\Vert"\geq\Vert|T$$x\in \mathcal{H}$$|^{t}x\Vert^{\epsilon+t}$

We also remark that the classes of paranormal and $absolute-(1,1)$ -paranormal operators are the same. These operator classes are defined by norm inequalities. Inclusion
relations among above operator classes are well known [12] as follows: For a fixed$p>0$ ,(1.1)$\{p-hyponormal\}\subset$$\subset${ –quasihyponormal}class{ $absolute-(p,$ -paranormal}.
$p$$A(p, 1)\subset$$1)$

The above inclusion relations are all proper. To study these operators, weighted shiftoperators are very usefull tools. As an extension of weighted shift, weighted composition operators (it’s definition will be introduced in the below) are known. Hence, tostudy some operator classes related to hyponormal operators, it is better that we know
properties of weighted composition operators. Study of hyp,onormal weighted comp&sition operators has been started by A. Lambert in [7]. Recently, weak hyponormalcomposition operators are studied in $[2, 1]$ they have shown some characterizations of
weak hyponormal composition operators, and obtained concrete examples for related
hyponormal composition $6perators$ . But they discussed on composition operators (notweighted composition operators), mainly.
In this report, we shall obtain some characterizations of related hyponormal weightedcomposition operators. In section 2, we shaIl prepare the definition and baeic properties of weighted composition operators. In section 3, we shall discuss acharacterization of and -hyponormalities of weighted composition operators. They areextensions of the results in [7] and [2]. In section 4, we shall show that the claes\’eof -quasihyponormal and $absolute-(p, 1)$ -paranormal weighted composition operators
are the same. This is an extension of the results in [1].
)

$p$

$p$

$\infty$20
2. DEFINITION AND BASIC PROPERTIES OF WEIGHTED COMPOSITION OPERATORSIn this section, we shall introduce the definition and basic properties of weighted
composition operators.

measure space. A measurable transforand $\mu\circ T^{-1}\ll\mu$ . For a non negativewithason, define the weighted composition operator

Definition 4. Let: $X$
mation$T$$arrow$$w\in L^{\infty}(X, \mathcal{F}, \mu)$$(X,\mathcal{F}, \mu)$

$X$

be a

$\sigma- finite$$T^{-1}\mathcal{F}\subseteq \mathcal{F}$

$C$

for

$Cf=wf\circ T$

Especially, the case$w\equiv 1$$f\in L^{2}(X,\mathcal{F}, \mu)$$L^{2}(X, \mathcal{F}, \mu)$.

, we call $C$ a composition operator, simply.

$C$is bounded. We can consider that weighted
In the casecomposition operators ar kind of shift operators.
$h= \frac{d\mu oT^{-1}}{d\mu}\in L^{\infty},$$e$

Example 2.1. Let $X=N$ , a transformation $T(n)$ be$T(n)=\{\begin{array}{ll}1 (n=1)n-1 (n\geq 2)\end{array}$and $w=(O, 1,2,2, \cdots)\in l^{\infty}$ . Then for $f=(f_{1}, f_{2}, \cdots)\in l^{2}$ ,$Cf=wf\circ T=w(fi, f_{1}, f_{2}, f_{3}, \cdots)=(0, fi, 2f_{2},2f_{3)}\cdots)$,i.e., $C$ is weighted shift which is the same as $U$ in Example 1.1.

.
Let $Ef=E(f|T^{-1}\mathcal{F})$ be the conditional expectation of with respect toon the average with respect to$Ef$ derived its uses from the idea that it represents$\int_{A}fd\mu=\int_{A}Efd\mu$ . This means that except. Specifically, for each-measurable, $Ef$ and are never related by a pointwise inequality,
when isand conditional expectation is of limited value in making pointwise estimates to thevalue of a function.
$f$$T^{-1}\mathcal{F}$$f$.

$A\in T^{-1}\mathcal{F},$$T^{-1}\mathcal{F}$$f$$f$$T^{-1}\mathcal{F}$

Example 2.2. Let $X=N$ , a transfornation $T(n)$ be$T(n)=\{\begin{array}{ll}1 (n=1)n-1 (n\geq 2).\end{array}$

Then$T^{-1}\mathcal{F}$is generated by the atoms{1, 2}, {3}, {4},Moreover the measure,$\cdots$.

is defined by $\mu(\{n\})=1$ for $n\in$ N.
$\cdot$$\mu$

Hence for $f=$$(f_{1}, f_{2}, f_{3}, \cdots)$$Ef=( \frac{\mu(\{1\})f_{1}+\mu(\{2\})f_{2}}{\mu(\{1,2\})}, \frac{\mu(\{1\})f_{1}+\mu(\{2\})f_{2}}{\mu(\{1,2\})}, \frac{\mu(\{3\})f_{3}}{\mu(\{3\})}, \frac{\mu(f^{\backslash }4\})f_{4}}{\mu(\{4\})}, \cdots)$$=( \frac{f_{1}+f_{2}}{2}, \frac{f_{1}+f_{2}}{2}, f_{3}, f_{4}, \cdots)$.

To study weighted composition operators, we prepare some important propertiesof conditional expectation as follows: First of all we note that for any nonnegative21function $f,$ $supp(Ef)$ is the smallest (up to null sets)[9], i.e.,$T^{-1}\mathcal{F}$set containing $supp(f)$ in$supp(f)\subseteq s.upp(Ef)$always holds. Hence,$\frac{f}{Ef}$is well defined.

Lemma A. Let $C=U|C|$ be the polar decomposition of weighted composition operator$C$and$h= \frac{d\mu oT^{-1}}{d\mu}$. Then for $f\in L^{2}(X, \mathcal{F}, \mu)$ , the following hold:(i) C’ $f=hE(wf)\circ T^{-1}[7]$ ,(ii) $C^{*}Cf=hEw^{2}oT^{-1}f$ .
In what follows, we write the modulus of $C$ by $a,$ $i.e.,$ $C^{*}Cf=hEw^{2}\circ T^{-1}f=a^{2}f$ .

Moreover by $hoT>0[6]$ , the Panial $isomet\eta$ part

$U$of can be erpressed as follows:$C$.

$Uf= \frac{w}{a\circ T}f\circ T$$i.e.$ ,Lemma $B$ ([11]). The conditional expectation $E$ is a projection onto
if
then there exists
such that $Eg=G\circ T.$ Hence$E(g\circ T\cdot k)=g\circ T\cdot Ek$ and $E(Eg\cdot k)=Eg\cdot Ek$ hold.
More properties are listed in [2].
$T^{-1}\mathcal{F},$$g,$ $k\in L^{2}(X, \mathcal{F}, \mu)_{f}$$G\in L^{2}(X, \mathcal{F}, \mu)$3. CHARACTERIZATIONS OF

$p$

AND

$\infty$-HYPONORMAL OPERATORSIn this section, we shall introduce characterizations of andcomposition operators.
$p$$\infty$-hyponormal weightedTheorem 3.1. For $p>0_{f}$ a weighted composition operator $C$ is p-hypono rmal if andonly if the following conditions hold:(i) $supp(w)\subseteq supp(a)$ ,(ii) $E[( \frac{aoT}{a})^{2p}\frac{w^{2}}{Ew^{2}}]\leq 1a.e$ .

The case $w\equiv 1$ has been shown in [2]. To prove Theorem 3.1, we use the following
characterization of hyponormal weighted composition operators:([7]). A weighted composition operator
Theoremthe following conditions hold:(i) $supp(w)\subseteq supp(a)$ ,$C$$C$

is hyponomal if and only if(ii) $E[( \frac{aoT}{a})^{2}\frac{w^{2}}{Ew^{2}}]\leq 1a.e$.

Proof of Theorem 3.1. Let $C=U|C|$ be the polar decomposition of C.

hyponormal if and only if $C_{p}=U|C|^{p}$ is hyponormal.

position operator as follows:$C_{p}$is Pweighted
comis also a$C$$C_{p}f=U|C|^{p}f=Ua^{p}f= \frac{w}{aoT}a^{p}\circ Tf\circ T=wa^{p-1}\circ Tf\circ T=w_{p}f\circ T$,

22where $w_{p}\equiv wa^{p-1}oT$ . Then by Lemma
(3.1)

$B$,

$Ew_{p}^{2}=E(w^{2}\cdot a^{2(p-1)}oT)=Ew^{2}\cdot a^{2(p-1)}oT$

Moreover we have $|C_{p}|f=|C|^{p}f=a^{p}f$ . Hence by Theorem$C$.

, we have only to provethat(i) $supp(w_{p})\subseteq supp(a^{p})\Leftrightarrow supp(w)\subseteq supp(a)$ ,(ii),$E[( \frac{a^{p}\circ T}{a^{p}})^{2}\frac{w_{p}^{2}}{Ew_{p}^{2}}]\leq 1\Leftrightarrow E[(\frac{a\circ T}{a})^{2p}\frac{w^{2}}{Ew^{2}}]\leq 1$

see [CY].

口

For each non-negative
inequality andis increasing on $p>0[8]$ by Holder’s. Then there exists$f\in L^{2}(X, \mathcal{F}, \mu),$$(Ef^{p})^{\frac{1}{p}}\leq\Vert f\Vert_{\infty}<+\infty$$(Ef^{p})^{\frac{1}{p}}$$M(f)=s- \lim_{parrow\infty}(Ef^{p})^{\frac{1}{\rho}}$and we call it minimal majorant of $f$ . It is known that $f.\leq M(f)$ holds in [8].

Next we will show a characterization of oo-hyponormal weighted composition operators as follows:
Theorem 3.2. A weighted composition operator $C$ isthe following conditions hold:.
(i) $supp(w)\subseteq supp(a)$ ,(ii) a $oT\leq a$ on $x_{\sup p(w)}$ ,
wheremeans the characteristic function on $N$ .

$\infty$-hyponormal if and only if$\chi_{N}$To prove Theorem 3.2, we shall prepare the following lemma:
Lemma 3.3. Let$a,$ $b\in L^{2}(X,\mathcal{F},\mu)$s-with$a,$ $b\geq 0$. Then$\lim_{parrow\infty}\{E(a^{p}b)\}^{\frac{1}{p}}=M(a\chi_{\sup p(b)})$.

Proof is given in [CY].

Proof of Theorem 3.2. By the definition of $\infty hyponormality$ of$C,$ $C$

is P-hyponormalfor all $p>0$ , that is, $supp(w)\subseteq supp(a)$ and$E[( \frac{a\circ T}{a})^{2p}\frac{w^{2}}{Ew^{2}}]\leq 1$

hold for all $p>0$ by Theorem 3.1. Then by Lemma 3.3 and $f\leq M(f)$ ,$1 \geq s-\varliminf_{\rangle p\infty}(E[(\frac{a\circ T}{a})^{2p}\frac{w^{2}}{Ew^{2}}])^{\frac{1}{p}}$$=M(( \frac{aoT}{a})^{2}\cdot x_{\sup p())}Bw^{2}\neg w\geq(\frac{a\circ T}{a})^{2}\cdot x_{\sup p()}B\neg ww^{2}\cdot$

Here, by $supp(w^{2})\subseteq supp(Ew^{2})$ , we have a $oT\chi_{\sup p(w)}\leq a$ .
$supp(w)\subseteq supp(a)$ we have (ii).

Moreover, by23
Conversely, if$a$$oT\leq a$on $supp(w)$ holds, then we have

$1 \geq Ew^{2}\cdot\frac{1}{Ew^{2}}=E(\frac{w^{2}}{Ew^{2}})\geq E[(\frac{a\circ T}{a}\cdot x_{\sup p(w)})^{2p}\frac{w^{2}}{Ew^{2}}]=E[(\frac{a\circ T}{a})^{2p}\frac{w^{2}}{Ew^{2}}]$hold for all $p>0_{r}$ that is, $C$ is$\infty$-hyponormal.

$\square$

4. EQUIVALENT CLASSES

In this section, we shall prove that for a fixed $p>0$ , the class of p.quasihyponormal
weighted composition operators coincides with the class of $absolute-(p, 1)$ -paranormal
weighted composition operators, and obtain their characterization. Hence some weakhyponormal classes in (1.1) are coincide with each other in the weighted composition
operators case. We remark that, $absolute-(p, 1)$ -paranormal has been introduced in
the name absolute-p.paranormal in [5], firstly.

Theorem 4.1. Let $C$ be a weighted composition operator. For $p>0$ , the following
conditions are equivalent:(i) $C$ is -quasihyponormal,(ii) $C$ belongs to class $A(p, 1)$ ,$p$(iii) $C$ is $absolute-(p, 1)$ -paranormal,(iv) $a^{2p}oT\cdot Ew^{2}\leq E(w^{2}a^{2p})$ .

The case $w\equiv 1$ has been already shown in [1].

Proof. Inclusions (i)(ii)(iii) have been already known as (1.1). So wehave to show the inclusions (iv)(iv).
(i) and (iii)$C$(iv)
Proof of(i).
is -quasihyponormal if and only if$\Rightarrow$$\Rightarrow$$\Leftrightarrow$$\Leftrightarrow^{.}$$\Leftrightarrow$$p$.

$C^{*}(C^{*}C)^{p}C\geq C^{*}(CC^{*})^{p}C$For$f\in L^{2}(X, \mathcal{F}, \mu)$, by simple calculation (detail is given in [CY]), we have.

$C^{*}(C^{*}C)^{p}Cf=hE(w^{2}a^{2p})\circ T^{-1}f$

On the other hand,$C^{*}(CC^{*})^{p}Cf=(C^{*}C)^{p+1}f=a^{2(p+1)}f$

Hence

$C$.

is -quasihyponormal if and only if$p$(4.1)$a^{2(p+1)}\leq h(Ew^{2}a^{2p})\circ T^{-1}$$\Leftrightarrow a^{2(p+1)}\circ T\leq h\circ TE(w^{2}a^{2p})$$\Leftrightarrow a^{2p}o$$\Leftrightarrow(iv)$

TEw $2\leq E(w^{2}a^{2p})$.

Proof of (iii)(iv). In [4, Section 3.5.5, Theorem 1],
paranormal if and only if$\Leftrightarrow$(4.2)$C^{*}|C|^{2p}C-(p+1)\lambda^{p}|C|^{2}+p\lambda^{p+1}I\geq 0$Here(1) $C^{*}|C|^{2p}Cf=hE(w^{2}a^{2p})oT^{-1}f$ ,for all$C$

is $absolute-(p, 1)-$$\lambda>0$.

24(2) $|C|^{2}f=a^{2}f$ .
Then (4.2) is equivalent to$hE(w^{2}a^{2p})\circ T^{-1}-(p+1)\lambda^{p}a^{2}+p\lambda^{p+1}\geq 0$

Put$\lambda=a^{2}$for all$\lambda>0$.

, then it is equivalent to$hE(w^{2}a^{2p})\circ T^{-1}\geq a^{2(p+1)}$,

and it is equivalent to (4.1), so does (iv).

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DEPARTMENT OF MATHEMATICS, KANAGAWA UNIVERSITY, YOKOHAMA 221-8686, JAPANE-mail address: chiyomOlQkanagawa-u. ac. jp
DEPARTMENT OF MATHEMATICS, KANAGAWA UNIVERSITY, YOKOHAMA 221-8686, JAPANE-mail address: $yamazt26Qkanagawa-u$ . ac. jp