Contemporary Mathematics

Volume 232, 1999

Multiplication and Composition Operators
between Two Lp -Spaces

Hiroyuki Takagi and Katsuhiko Yokouchi

Abstract. In this paper, we determine the symbol functions that induce mul-
tiplication operators between two Lp-spaces and the symbol transformations
that induce composition operators between them. Also, we characterize those
operators that have closed ranges.

Introduction

Let (X, ЯЛ, p) be a а-finite measure space. By L(X), we denote the linear space
of all (equivalence classes of) DJI-measurable functions on X, where we identify any
two functions that are equal //-almost everywhere on X. For 1 < p < oo, we
abbreviate the Lp-space LP(X, 9R, /t) to LP(X), and denote its norm by || • ||Lp(x)-
Take a function и in L(X). Then и induces a linear operator Mu from LP(X) into
L(X) defined by

Muf(x) = u(x) /(z)	(X e X, f e LP(X)).

If Mu takes LP(X) into Lq(X), then we call Mu a multiplication operator from
LP(X) into L9(X) (1 < q < oo). The study of multiplication operators has a very
long history. From books on functional analysis, we can learn many properties of
the multiplication operators on various function spaces, including Lp-spaces. But
there has been relatively little study of the operators between two different Lp-
spaces. In this paper, we consider such operators. In Section 1, we determine
the symbol funciton и that induces a multiplication operator Mu from LP(X) into
Lq(X) (Theorems 1.2, 1.3 and 1.4). In Section 2, we characterize the multiplication
operators from LP(X) into L9(X) that have closed ranges (Theorems 2.3, 2.4 and
2.5). The results in case p q are apparently new. So, we present the explicit
statements and give their proofs. This material will then be applied to composition
operators.

Next, let (Y, 91, v) be another а-finite measure space. Similarly, we use the
symbols L(Y) and LP(Y) to denote the linear space of all 9t-measurable functions on
Y and the Lp-space Lp(Y,9t, i/), respectively. Let be a meaurable transformation
from Y into X, namely, a mapping from Y into X such that E € 9R implies
€ 91. If i/(</?-1(E)) = 0 for all E € 9R with p(E) — 0, then </? is said to be

1991 Mathematics Subject Classification. Primary 47B38.
nonsingular. For any nonsingular measurable transformation <p from Y into X,
induces a linear operator Cv from LP(X) into L(Y) defined by

C„№) = ЛЛ)) luer, /e L”(X)).

Here, the nonsingularity of <p guarantees that is well defined as a mapping
of equivalence classes of functions. If maps LP(X) into Lq(Y), then we call
a composition operator from LP(X) into Lq(Y). Topics involving composition
operators have received a great deal of attention in recent years. Many of these
topics can be found in the reports [5,13] or the recent monographs [6,18,24,30]. In
particular, composition operators on Lp-spaces are studied by R.K. Singh [19-23],
A.Kumar [11], A.Lambert [10,12], W.C.Ridge [14], R. Whitley [27], J.W.Carlson
[2], W. Feldman [7] and many other mathematicians. They studied the operators
from Lp-space (especially, L2-space) into itself. In this paper, we deal with the
composition operators between different Lp-spaces. In Section 3, we completely
determine the symbol transformation p that induces a composition operator
from LP(X) into Lq(Y) (Theorems 3.4, 3.5 and 3.8). In Section 4, we characterize
ones that have closed ranges (Theorems 4.2, 4.4 and 4.5).

At this point, we give some preliminary comments on atoms. We are working
on a ст-finite measure space (X, DU, p). Take a set A 6 DU with p(A) > 0. We say
that A is an atom if, for any E € DU with E C A, we have either p(E) = 0 or
p(A \ E) — 0. Let A be an atom. Since p is ст-finite, it follows that p(A) < oo.
Also, every Oli-measurable function f (e L(X)) is constant ^-almost everywhere
on A. In fact, the constant a/ is given by otf —	f dp. From this point of

view, we can regard an atom as a point at which each function f e L(X) takes a
value af. For this reason, we adopt the notation

fW=^Lfdii

for each f 6 L(X) and each atom A.

Conversely, let us consider atom free sets. If an ЯЛ-measurable set E contains
no atoms, then we say that E is nonatomic. The nonatomic set has the following
remarkable property:

Proposition 0.1 (29 ; Exercise 10.11). Suppose that an ^Л-measurable set E
is nonatomic and that p(E) > 0. Then, for any number a with 0 < a < p(E), there
is an 'JR-measurable subset Ea of E such that p(Ea) — a.

Finally, we state the useful decomposition of a ст-finite measure space X.

Proposition 0.2 (29; Exercise 10.16). A а-finite measure space X is uniquely
decomposed as follows;

(0.1)	X=(uaJuB,

yneN J

where {An}neN is a countable collection of disjoint atoms and В is a nonatomic
set.

Throughout this paper, we use the decomposition (0.1). The representation
(0.1) is unique, when we identify any two sets whose symmetric difference is p-
measure zero. In this paper, we always handle Dll-measurable sets under this
identification. For instance, if E, F € DU, the equality E = F is interpreted as
ju((F \ F) U (F \ F)) = 0, and the inclusion F C F as /i(F \ F) = 0. For details,
see [8] and [29].

1.	Multiplication Operators

We begin with the following fundamental proposition, which follows as an easy
consequence of the closed graph theorem.

Proposition 1.1. Suppose 1 < p < oo and 1 < q < oo. Every multiplication
operator from LP(X} into Lq(X) is a bounded linear operator from LP(X) into
L^X).

We raise the most natural question in the present context:

Which function и induces a multiplication operator Mu from LP(X)
into Lq(X)?

In this section, we completely answer to this question. In light of Proposition 1.1,
the induced operator Mu is bounded, and so we can consider the operator norm
||Ми ||. We will give the norm estimate at the same time. In order to describe our
answer, it is convenient to divide an argument into three cases:

Case I p — q, Case II p> q, Case III p < q.

The result (Theorem 1.2) in Case I is well known. The Cases II and III are
considered by Axler [1], when X is an interval [—7г,7г]. Our results (Theorems 1.3
and 1.4) are inspired by his idea.

Case I	p = q

Theorem 1.2. Suppose 1 < p < oo and и G L(X). Then и induces a multipli-
cation operator Mu from LP(X) into LP(X) if and only if и G L°°(X). In this case,
Mu is a bounded linear operator from LP(X) into LP(X), and its norm is given by
l|MU|| = |Ы|£°О(Х)’

This theorem is well known. For the proof, refer to [4; Theorem II. 1.5, Example
III.2.2] or [9; Problem 64, 65].

Case II	p > q

Theorem 1.3. Suppose 1 < q < p < oo and и G L(X). Then и induces
a multiplication operator Mu from LP(X) into Lq(X) if and only if и G Lr(X),
where ~ = this case, Mu is a bounded linear operator from LP(X) into
Lq(X), and its norm is given by ||Mu|| = IMI^r^.

Proof, “if” part. Suppose и 6 Lr(X). For each f G LP(X), Holder’s in-
equality yields

- {C4|u|’5 ^У (Уl/l'” ^У У=
Hence Mu is a bounded linear operator (of course, a multiplication operator) from
LP(X) into Lq(X), and we have

(1-3)	||MU || < ||'W||£’"(X) •

“only if” part. Suppose that и induces a multiplication operator Mu from
17(X) into Lq(X). By Proposition 1.1, Mu is bounded and ||MU|| < oo. Define a
linear functional Ф on Lr (X), where 1	— 1, by

Ф(/)= [ fudp (JzLr\X)).

Jx

Then we see that Ф is bounded, as follows. Take f e Lr (X) arbitrarily. If | + = 1>
then + y and so, we can easily find two function fi € LP(X) and /2 6 Lq'(X)
such that

f=fi f2 and iiAiifP(x) = ii/2ii^(x) = ii/ii;'z(x) •

(For example, put /i(x) = |/(rr)|T and /2(2) — if f(x) / 0, and /i(x) = /2(2) = 0

/1 \x)

otherwise.) Using this factorization and Holder’s inequality, we compute

< IIм.II11/1 IL-w||/2||t,'(JQ = IIМиII ll/llLt(Jf|ll/llLt(x| = ЦЛМ \\f\\L,'m.

This shows that Ф is a bounded linear functional on Lr (X) with ||Ф|| < ||MU||. We
now apply the Riesz representation theorem to obtain a function v e Lr(X) such
that

*(/)= I fvdfi	and ||Ф|| = ||<.m .

Then we have

[ fudp = [ fvdp (feLr'(X)),

Jx Jx

which forces that и = v (^-almost everywhere on X). Hence we arrive at и e £r(X),
and have

(1-4)	ll<.(x) = ll<-m = 11*11 < l|M„||.

Finally, the equality ||Ми|| = ||u||Lr(X) follows from (1.3) and (1.4).	□

Case III p < q In the proof of our theorem in this case, we use the decom-
position (0.1) of X.

Theorem 1.4. Suppose 1 < p < q < 00 and и 6 L(X). Then и induces a
multiplication operator Mu from LP(X) into Lq(X) if and only if и satisfies the
following two conditions:

(1.5)	u(x) — 0 for p-almost all x E B.

(1.6)	sup —< co , where 1 4-1 = 1.

g(An)	d s p
In this case, Mu is a bounded linear operator from LP(X) into Lq(X), and its norm
is given by ||MU|| = sup 1Ц^П^ .

neN p(An)s

For the proof, we need the following elementary fact. The related results are
obtained in [15].

Lemma 1.5. Suppose 1 < p < q < oo. If an ^Я-measurable set E is nonatomic
and if pfE) > 0, then there exists a function fa e LP(X) such that / |/o|9 dp = oo-
Je

Proof. Choose a number a so that 0 < a < p(E). Since E is nonatomic, we
repeatedly use Proposition 0.1 to find a sequence Ei,E?,- • • e 9Л of disjoint subsets
of E such that p(Ek) = for к = 1,2,.... We define a function /о on X by

oo 1

f° ~ S i XE >
к=1 p(Ek)«

where xEk is the characteristic function of Ek- Then fa becomes the desired func-
tion, because

/ \fo\pdp = / 12	dp = S	’ = a ’ S (2)	< °0’

Jx	Jxk=ifi(Ek)i k fe=1	fc=ivz/

while

Г	Г oo	1	oo

/	d^=	£ ХЕк^ = 12 1 = °° •	□

Je	JE fc=l K'	k=l

Proof of Theorem 1.4. “if” part. Suppose that both (1.5) and (1.6) hold.

IttlA IIs

Put b — sup — n . Then, for each f e LP(X), we have
nEN Д(-^п)

\\M„f\\l,,„ = [	IWT-1M+ S [ WI’rfM

1 J Jx	J В	nEN JAn

= 0+ S |И(Л„)|’|/(Л„)|’м(Л„)

neN

(17)	= E	l/(-4~)l’M(^)1+!

nEN V	/

< ы S 1ЛЛ„)|МА.); = Ы s (|/(A„)|”M(A„))?.

If ll/b(x, < 1, then |/(An)|»/r(An) = / 1/1»dp < J 1/1» dp = ||/||fP(J() < 1 for
all n e N, and so, using > 1, we have

S (|/(A„)|»M(A„))? < E |/(Л„)|»м(Л„)
n€N	n€N

= e [ \f\pdp< [ i/i» dp = ил;•	< i.

n€NJAn	Jx	1 }

Substituting (1.8) into (1.7), we get ||Mu/||Lq(X) < b° for all f e LP(X) with
II/IIlp(x) — 1- Hence Mu is a bounded linear operator (of course, a multiplication
operator) from LP(X) into Lq(X), and

||MU||<^

(1.9)

|«(An)|
= SUp —1-----j-

n&N р(Ап)з
“only if” part. Suppose that и induces a multiplication operator Mu from
LP(X) into Lq(X). By Proposition 1.1, Mu is bounded and ||MU|| < oo.

We first show (1.5). Assume that //({ж € В : и(ж) / 0}) > 0 . Then there
exists a positive number S such that //({ж € В : |гл(ж)| > £}) > 0. Put E={x£
В : |u(x)| > £}. Since E is nonatomic, Lemma 1.5 furnishes a function fa e LP(X)
such that /	= 00 • This function fa satisfies

Je

i	i

oo = W |/о|9^Г <{ [ W/o|’<M
к. «/ E	J к v E	✓

~	= II Mufo	(X) — 11Ш ||/o||L₽(x) < 0° ’

which is a contradiction. Hence we conclude that //({ж e В : u(x) / 0}) = 0. In
other words, u(s) — 0 for p.-almost all x e B.

Next, we examine the supremum in (1.6). For any n e N, put fn = —-—г Xa •
д(Ап)р

Then it is clear that fn e LP(X) and || fn||Lp(x) = T Hence we have

ИМ
ju(An)i ’

IIAfu|| — ||Mufn~ \ J lw^nl9^J
’fl	1’

[m(Ai)₽

Since this holds for any n e IV, it follows that

sup	< ||M„|| < oo ,

neN p\An)°

which is equivalent to (1.6). Together with (1-9), we obtain the norm estimate.

When the index set N in (0.1) is finite, the condition (1.6) in Theorem 1.4 is
automatically satisfied. In particular, if N is empty, then the theorem becomes:

Corollary. Suppose 1 < p < q < сю and X is nonatomic. Then the zero

operator is the only multiplication operator from LP(X) into Lq(X).

2. Multiplication operators with closed range

In this section, we investigate a multiplication operator Mu from LP(X) into
Lq(X) that has closed range. Here the statement “Mu : LP(X) —► Lq(X) has closed
range” means that the range MU(LP(X)) is closed in Lq(X). Thus, our problem is:

When does the multiplication operator have closed range?

As a preliminary, we consider the restriction space. Take a set G € 9И. If we
define a a-algebra 50^, in G and a measure pG on 50^, by

= {£Ж : E e ЯП} and pG(E) = p(E) (Ee^)>

then we obtain a new ст-finite measure space (G,5U^,/iG). Similarly, we use the
symbols L(G) and LP(G) to denote the linear space of all 91^.-measurable functions
on G and the Lp-space LP(G, SU^,/zG), respectively.
For u € L(X), u|g stands for the restriction of и to G. Clearly, u|g lies in L(G)
and induces the linear operator MU\G from LP(G) into L((7) such that

Mu\Gh(x) = u(x)h(x) (xeG, heLp(G)).

The next lemma summarizes the obvious relation between two operators Mu and
MU\g •

Lemma 2.1. Suppose 1 < p < oo and 1 < q < oo, and let Mu be a multipli-
cation operator from LP(X) into Lq(X}. Then MU\G is a multiplication operator
from LP(G) into Lq(G). If Mu : LP(X) —> Lq(X) has closed range, then MU\G :
LP(G) —► Lq(G) also has closed range. In addition, if G С {x e X . u(x) 0},
then MU\G is one-to-one.

For и e L(X), we write supp и for the support of u:

supp и — {x e X : и(ж) / 0}.

The support of и plays an interesting role in the discussion below, and we often
take it as the above set G. Though the next lemma is nothing but a straightforward
application of the inverse mapping theorem, it will make our first theorem easy to
prove.

Lemma 2.2. Suppose 1 < p < oo and 1 < q < oo, and let Mu be a multipli-
cation operator from LP(X) into Lq(X). Then Mu has closed range if and only if
there exists a constant c > 0 such that

(2.i)	II^u|s^IIl<j(s) > c II^IIlp(s) (h e LP(S)),

where S = supp u.

Proof. Notice that the range Mu(Lp(Xf) is contained in the space;

{ 6 Lq(X) : g(x) = 0 for ^-almost all x e X \	,

which is isometrically isomorphic to Lq(S). Hence A/U(LP(X)) is closed in Lq(X) if
and only if Mu\s(Lp(Sf) is closed in Lq(S). While, by Lemma 2.1, Mu\s is one-to-
one, that is, the kernel of Mu\s is {0}. From [4; Exercise III. 12.5] or [26; Theorem
IV.5.9], it follows that Mu|s(Lp(S)) is closed if and only if there exsits a c > 0
satisfying (2.1). Thus the lemma is proved.	□

With these preliminaries, we give a necessary and sufficient condition on и for
Mu to have closed range, which is our purpose.

Case I p = q The next theorem may be known. The proof of the case p — 2
is given by R.K. Singh and A. Kumar [21] or [4; Exercise II.2.16]. For the sake of
completeness, we here prove it in the general setting 1 < p < co.

Theorem 2.3. Suppose 1 < p < oo, and let Mu be a multiplication operator
from LP(X) into LP(X\ Then Mu has closed range if and only if there exists a
constant S > 0 such that |u(s) | > 6 for p-almost all x e supp u.

Proof. Write S — supp u.

“if” part. If |u(a:)| > S for ^-almost all x € S, then we have

II-^u|s^IIlp(s)

u\p\h\pdp
for all h e LP(S). Hence, Lemma 2.2 shows that Mu has closed range.

“only if” part. Suppose that Mu has closed range. According to Lemma 2.2,
we can choose a constant c > 0 satisfying (2.1). Take 6 — £, and put E — {x E S :
|u(s)| < 5}. Let us assume p(E) > 0. Since p is ст-finite, we find a set F e SUI such
that F С E and 0 < /z(F) < oo. Then the characteristic function xE lies in LP(S)
and satisfies

1 p

1Х/г|Р r IIXfIIlp(S) <C II^FIIlp(S) •

This is contrary to the choice of c. Hence we have p(E) — 0. In other words,
|u(x)| > 6 for jz-almost all x € S.	□

Case II p> q

Theorem 2.4. Suppose 1 < q < p < oo, and let Mu be a multiplication opera-
tor from LP(X} into Lq(X). Then the following assertions are equivalent:

(i) Mu has closed range.

(ii) Mu has finite rank.

(iii) u(x) = 0 for p-almost all x E B, and the set {n E N . u(An) / 0} is finite.

Proof. Put S = suppu and Nu = {n E N : u(An) / 0}. If p(S) = 0, then Mu
is the zero operator, and there is nothing to prove. So we assume that p(S) > 0.

(iii) => (ii). If (iii) holds, then S is expressed in the form:

S = U An = Ani U • • • U An/c for some positive integer k,
n&Nu

and the characteristic functions { y .	• • • y . } span the subspace

Ani 5	5 ™nk J

| g E Lq(X) : g(x) = 0 for ^.-almost all x E X \ S'} .

Since this fc-dimensional space contains the range MU(LP(X)), Mu has finite rank.

(ii) => (i). Any finite dimensional space is closed.

(i) => (iii). Suppose that Mu has closed range. First, we show that u(x) — 0 for
^-almost all x E B. Assume to the contrary that p({x E В : u(x) / 0}) > 0. Then
we have p({x E В : |u(rr)| > 5}) > 0 for some 5 > 0. Set G — {x E В : |u(a?)| > 5}
and define a function v on G by v(x} = for x E G. Then v induces a linear

J * (i) (ii) (iii) * v 7 u(x)

operator Mv from Lq(G) into L(G) as follows:

Mvg(x) = v(x) р(ж) (x E G, g E Lq(G)).

Now we will show that Mv maps Lq(G) into LP(G). Clearly, Mv is the inverse
operator of MU\G, and Lemma 2.1 says that MU\G is a bounded linear operator from
LP(G) into Lq(G) that has closed range. So, if we show that MU\G (Lp(Gf) = Lq(G),
then it follows that Mv maps Lq(G) into LP(G). Let us show MU\G (LP(G)) — Lq(G).
For any E E 9JL with p(E) < co, put hE(x) = —ttXpW for x E G. Then hE
u(x) л

belongs to LP(G), because

_L_ dp < — p(E) < co .

|tz|P	7

Moreover, Mu\GhE = xE, and so we have xE € Mu\g(Lp(G}). Hence the range
Mu|G (LP(G)) contains the subspace F of all linear combinations of such XE&- Here
we note that Mu|g(Lp(G)) is closed in Lq(G), and recall from [17; Theorem 3.13]
that F is dense in Lq{G). Then we obtain MU\G (LP(G)) = Lq(GT). Thus we see that
Mv maps Lq(G) into LP(G), that is, Mv is a multiplication operator from Lq(G)
into LP(G). Now, we can apply Theorem 1.4 (Corollary) to Mv, with и — v and
interchanging p and q. Noting that G is nonatomic, we get v(rr) = 0 for //-almost
all x e G. But this is impossible, because p(G) > 0 while v has no zero on G by the
definition of v. This contradiction shows that p({x e В : u(x) 0}) = 0 , which
means that u{x) — 0 for //-almost all x 6 B.

Next, we show that Nu is finite. Since we have shown that u(x) = 0 for //-almost
all x € B, we can write S = U An. Since we assume that //(S) > 0, it follows
neNu

that Nu / 0. This time, we put w(x) = —for x e S, and consider the operator
Mw. By replacing G and v in the preceding paragraph by S and w respectively, we
see that Mw maps Lq(S) into LP(S). Thus Mw becomes a multiplication operator
from Lq(S) into LP(S). Applying Theorem 1.4 to Mw, we get

1	|w(An)|r

sup . . . , ,--y-r-T — sup -----. . .	< 00 ,

neNu |u(An)|r//(An) neNu p(An)

where i + i — i. Now, put b = sup ,	/—7-3—7. Then b > 0, because Nu 0,

P r q	neNu |и(Ап)Г//(Ап)

and |u(An)|r//(An) > | for all n e Nu. While Theorem 1.3 says и 6 Lr(X). So,
we have

£ X - S |t/(An)|r//(An) = S [ \u\rdp< [ |u|rdp < oo .

n£Nu n€Nu	nENu JAn	JX

This implies that Nu is finite. Thus the implication (i) => (iii) is proved. □

Case III p < q

Theorem 2.5. Suppose 1 < p < q < oo, and let Mu be a multiplication opera-
tor from LP(X) into Lq(X). Then the following assertions are equivalent:

(i)	Mu has closed range.

(ii)	Mu has finite rank.

(iii)	The set {n G N : u(An) / 0} is finite.

Proof. Theorem 1.4 tells us that u(s) = 0 for //-almost all x e. B. So, the
implications (iii) => (ii) => (i) can be proved as well as the same parts of Theorem
2.4. Thus only the implication (i) =Ф (iii) is to be proved. Suppose that Mu has
closed range. We must show that the set Nu = {n e N : u(An) / 0} is finite. We
exclude the trivial case Nu = 0 and assume Nu / 0. If we put S = supp u, then we
can write S = U An. Define a function w on S by w(x) = for x € S. As
neNu	u{x)

in the proof of Theorem 2.4, we can show that w induces a multiplication operator

Mw from Lq(S) into LP(S). This time, we appeal to Theorem 1.3, and we obtain
w G LS(S), where 1 + 1 = 1. While Theorem 1.4 says that b = sup	< oo.

q s p	neN P\An)

Since Nu / 0 implies b > 0 and since |w(An)|s//(An) =
it follows that

,	| for all n G M

S l< S |w(An)|s//(An) = s
neNu° n€Nu	neNu

/ |w|sd// — I |w|sdps < oo ,
J An	Js
which shows that Nu is finite.

□

3.	Composition operators

We are now in a position to study composition operators. We begin with an
analogue of Proposition 1.1.

Proposition 3.1. Suppose 1 < p < oo and 1 < q < oo. Every composition
operator Cp from LP(X) into Lq(Y) is a bounded linear operator from LP(X} into

From now on, our discussion is along the line of the argument developed in
[25]. For a nonsingular measurable transformation from Y into X, we define a
measure vipr1 on 9Л as

= i/(<p_1(E)) (ЕеШГ).

Then the nonsingularity of <p means that is absolutely continuous with respect
to p. Hence the Radon-Nikodym theorem ensures the existence of a nonnegative
function Up 6 L(X) such that

(3.1)	/ Up dp (EtWl).

Je

This function Up links composition operators to multiplication operators.

Lemma 3.2. Suppose 1 < p < oo and 1 < q < oo, and let be a nonsingular
measurable transformation from Y into X. Then we have

II ^v/llb’ (У) = II M^f IIl« (X)
for all f e LP(X).

Proof. Using the formula in [8; Theorem 39.D] or [16; Problem 15.6], we have

\\c*f\\qL4Y)

for all f e LP(X).

Lemma 3.2 gives us the following helpful information:

Lemma 3.3. Suppose 1 < p < oo and 1 < q < oo, and let be a nonsingular
measurable transformation from Y into X. Then <p induces a composition operator
Cp from LP(X) into Lq(Y) if and only if rfpf, induces a multiplication operator
from LP(X) into Lq(X). In this case, we have ЦС^Ц = 1111.

Here the next questions come up naturally:

Which transformation ip induces a composition operators Cp from LP(X)
into Lq(Y)?

What is the norm of Cp?

As before, we consider three cases:

Case I p — q, Case II p > q, Case III p < q.

Case I p = q The answer in this case is essentially due to R.K. Singh [20]
(see also [13] or [24]). His original theorem is stated in the setting X = Y, but it
remains true, with the same proof, even if we generalize the statement as follows:
(3-4)

= inf <

Theorem 3.4 (R.K. Singh [20]). Suppose 1 < p < сю, and let p be a nonsin-
gular measurable transformation from Y into X. Then the following assertions are
equivalent:

(i)	p induces a composition operator from L^tX) into IFfY).

(ii)	belongs to L°°(X).

(iii)	There exists a constant b such that vprx(E) < bp(E) for all E 6 ЯП.

If one of the above assertions holds, is a bounded linear operator from LP(X)
into LFfY) and its norm is given by

(3.2)	||Cv||>’ = ||uv||b>(x)=inf{b>0 :	(BeWt)}.

Case II p> q For any F 6 9Л, we put

Qy(F) = inf (b > 0 : vip"* 1 (ii) (iii) (E) < b p(E) (F еЯЛ,Е C F) } .

Then the right side of (3.2) is written as Qlfi(X). Also, it is obvious that

(3.3)	up~\E) < QpfF) p(E) (E € ЯИ, E C F).

Theorem 3.5. Suppose 1 < q < p < oo and I + i = |. Let p be a nonsin-
gular measurable transformation from Y into X. Then the following assertions are
equivalent:

(i) p induces a composition operator from LP(X} into L4(Y).

(ii) Uy belongs toLv(X).

OO	r

(iii) There exists a partition {Fj}°lx of X such that £	p(Fj) < oo.

J=i

If one of the above assertions holds, Cv is a bounded linear operator from LP(X)
into Lq(Y) and its norm is given by

: {Fj} is a partition of X >

J=i	7

We will need two lemmas.

Lemma 3.6. For any F e ЯЛ, we have Qtp(F) — ess sup Uy (ж).

xeF

Proof. For any E € ЯЯ with E C F, we have

i/p~1(E) = / u^dp< (ess sup Uy (ж)) p(E) < (ess sup Uy (ж)) /z(F) .

Hence Qy(F) < ess sup Uy (ж). For the opposite inequality; Qy(F) > ess sup Uy (ж),
xEF	xEF

it is enough to consider only the case Qy(F) < oo. Let us assume that

p({x e F : иу(ж) > Qy(F)}) > 0 .

Then there is a 8 > 0 such that p({x e F : иу(ж) > Qy(F) + <5}) > 0. Since p is
ст-finite, we find a set E € 9Л such that E С {ж € F : u^x) > QtptF) + 5} and
0 < p(E) < oo. From (3.3) and (3.1), it follows that

Qy(F)Ju(F)>u(/?-1(F)= [ uvdp>(Qv(F) + 8)p(E),

JE
which is impossible. This contradiction shows that p({x 6 F : u^(x) > Q^(F)}) —
0 , that is, Q<p{F) > ess sup (ж), completing the proof.	□

xeF

Lemma 3.7. fxu^ dp — inf	{^j} is ° partition of X }

Proof. Denote the infimum of the right side by 1^. For any partition {Fj} of
X, we use Lemma 3.6 to have

/ u^dp= £ / u^ dp < £ (ess sup	(ж)) ’ p(Fj) = £	p(Fj).

JX	j=lJFj	j=l' xeFj	'	j = l

Hence we have / dp < 1^. To verify the opposite inequality, choose a number
Jx

a > 1 arbitrarily, and set

Gm = {x e X : a™-1 < «^(ж)’ < am] .

for each integer m. If {F^}^ is a rearrangement of	and {ж € X :

u^(x) = 0}, then {Fj} clearly becomes a partition of X. Moreover, we use Lemma
3.6 to see

<	22 <?(ip(Fj)«ju(Fj) = £ (ess sup (ж) Vjz(Fj)
J = 1	J = 1 V xEFj	'

= § (esssupu^H< § a™ p(Gm) - a § a™-1 p(Gm)
m= — oo4 xEGm	z	m= — oo	m= — oo

<	a 22 I u<pi dp — a 22 I u<pi dp — a I u,^ dp.

m= — ooJGrn	j = l JFj	JX

Since this holds for any a > 1, we obtain Iv < / dp. Thus the lemma is
Jx

proved.	□

Proof of Theorem 3.5. (i)	(ii). By Lemma 3.3, (i) holds if and only if

induces a multiplication operator from LP(X) into Lq(X), and we have

ЦС^Ц — ||M^||. Also, Theorem 1.3 tells us that induces a multiplication
operator from LP(X) into Lq(X) if and only if 6 Lr(X), and that
II^Wi = ii^Hlr(X). Since e Lr(X) precisely when u^ 6 L’(X), we
establish the equivalence of (i) and (ii). At the same time, we get

(3.5)	||C<p|| = II |1ьг(Х) •

(ii)	<=> (iii). As before, denote the infimum in Lemma 3.7 by 1^. Since uv G

(X) means / u^ dp < oo, Lemma 3.7 shows that (ii) is equivalent to < oo.
Jx

Moreover, we can easily seen that (iii) holds if and only if 1^ < oo. Thus we obtain
the equivalence of (ii) and (iii).

Finally, if one of assertions (i) - (iii) holds, then the above equivalences make (i)
valid, so Proposition 3.1 shows that is a bounded linear operator from LP(X)
into L9(X). Also, the norm estimate (3.4) follows from (3.5) and Lemma 3.7.	□

Case III p <q

Theorem 3.8. Suppose 1 < p < q < oo, and let <p be a nonsingular measurable
transformation from Y into X. Then the following assertions are equivalent:
(i)	9? induces a composition operator from LP(X) into Lq(Y).

и (An)p

(ii)	u^fx) — 0 for p-almost all x G B, and sup —-—< сю.

nEN p(An)4

(iii)	— 0, and there is a constant b such that г/<р~1(Ап)р < bp(An)9

for all n e N.

If one of the above assertions holds, Cv is a bounded linear operator from LP(X)
into Lq(Y) and its norm is given by

ncyi1” =sup„w

(3.6)	f	.

= inf I b > 0 : 1Л/?-1(Ап)р < bp(An)q (ne Лг) | .

Proof, (i) о (ii). This equivalence is an immediate consequence of Lemma
3.3 and Theorem 1.4, because

(z---------s \ 22 z ----------\ pq

м(Л„) ) ~ ( М(Л„)1 )	h

where |	| = i. At the same time, we obtain the first equality in (3.6).

(ii)	4Ф (iii). Substitution E = В in (3.1) gives

v<p~1 (B) = / Uy dp.

J в

Noting that is nonnegative, we see that и^(х) = 0 for ^-almost all x € В if and
only if \B) = 0. Next, we use the equality

(3.7)	(A^) = I Utp dp — tz^(An) p(An) (n G AT)

J An

to compute

Ц<ДАП)Р _ v<p г(Ап)р
м(Л„)’-’“ g(A„)’	'

While a routine work shows that

sup	= inf { b > 0 : i/<p-1(An)p < bp(An)q (n G AT) | .

neN P\An)	l	)

Consequently, we obtain the second equality in (3.6), which shows that

и (An)p

sup ———< oo if and only if there is a constant b such that vtp~l(An)p <
nCN p(An)

bp(An)q for all n € N. Thus we establish the equivalence of (ii) and (iii). □

Theorem 3.8 has the following corollary:

Corollary. Suppose 1 < p < q < oo. If X is nonatomic and v(Y) > 0, then
there is no composition operator from LP(X) into Lq(Y).

Proof. Assume that there exists a composition operator from LP(X) into
Lq(Y). Then Theorem 3.8 shows that v<p~x(B) = 0. Since X is nonatomic, namely
X = B, it follows that </?-1(B) — <p-1(X) = Y. Hence v(Y) — 0, which is contrary
to hypothesis.	□
4.	Composition operators with closed range

The problem in this section is:

When does the composition operator have closed range?

In [3], W.A. Cima, J. Thomson and W. Wogen settled this problem for composition
operators on Hardy space. Later, R.K. Singh and A. Kumar [21] proved the similar
result for composition operators on L2-space. In this section, we try to extend
the latter result to our setting. Explicitly speaking, for a composition operator Cp
from LP(X) into £9(У), we give a necessary and sufficient condition for Cp to have
closed range. Here the statement “C^ : LP(X) —> Lq(Y) has closed range” means
that the range Cp(Lp(X')') is closed in Lq(Y).

The next lemma is a basic tool in this section.

Lemma 4.1. Suppose 1 < p < oo and, 1 < q < oo, and let Cp be a composition
operator from LP(X) into Lq(Y). Then Cp has closed range if and only if the
multiplication operator from LP(X) into Lq(X) has closed range.

Proof. In Lemma 3.3, we have seen that is a bounded linear oper-
ator from LP(X) into Lq(X). Suppose that M^j^{Lp{X')') is closed in Lq(X).
Choose {fn} C LP(X) and g E Lq(Y) so that ||C^/n — </||L<j(y) —> 0. Then
{Cpfn} is a Cauchy sequence, and so it follows from Lemma 3.2 that {M^^fn}
is a Cauchy sequence in Lq(X). Since Af^u^(Lp(X)) is complete, we find an
f e LP(X) such that \\M^^fn —	» 0. Then Lemma 3.2 shows

that \\Cpfn-Cpf\\Lq(Y) ~* 0- Hence we get g = Cpf G C<p(Lp(X)). Consequently,
we see that Cp(Lp(X)) is closed in Lq(Y). Thus the “if” part is proved. The “only
if” part can be proved by interchanging the roles of Cp and	□

Case I p — q To state the theorem, we must mention the collection £p of all
E G 9Л which satisfy the following two conditions:

(4.1)	pfE) < co.

(4.2)	If F E 9JI, F С E and ^-1(Е) = 0, then p(F) = 0.

Theorem 4.2. Suppose 1 < p < oo, and let Cp be a composition operator from
LP(X) into LP(Y). Then the following assertions are equivalent:

(i)	Cp has closed range.

(ii)	There exists a constant 6 > 0 such that Up(x) > 6 for p-almost all x E
supp Up.

(iii)	There exists a constant c > 0 such that v(p~x{E) > cju(E) for all E G £p.

We first make a remark on the collection £p.

Lemma 4.3. Suppose E G and p(E) < oo. Then E G £p if and only if
E C supp Up.

Proof, “if” part. Suppose that E C suppu^,. Let F G 9Л satisfy FcE and
i/</?-1(F) = 0. Since F C supp Up, we can write F = {x E F : Up(x) > 0}. Hence,
if ju(F) > 0, then there is a 6 > 0 such that p({x E F : Up(x) > <5}) > 0, and so,
putting G = {x E F -. Up(x) > 6}, ~we reach an impossible inequality:

0 = iAp-1(F) = I Updp> I Up dp > 6 p(G) > 0 .

Jf Jg

As a consequence, we obtain p(F) = 0. Thus E satisfies (4.2), and E E £p.
“only if” part. Suppose that E E 8p. Put F = E’\suppu^>. Then F С E, and
since Up(x) — 0 for //-almost all x e F, it follows that i/<p-1(F) = j Up dp — 0.
Hence (4.2) yields p(F) = 0, which means E C supp Up.	□

Proof of Theorem 4.2.	(i) <=> (ii). By Lemma 4.1, (i) is equivalent to

the fact that the multiplication operator from LP(X) into LP(X) has closed
range. Also, Theorem 2.3 tells us that : LP(X) —► LP(X) has closed range if
and only if there exists a constant 8q > 0 such that ^/up(x) > 8q for //-almost all
x e supp The last condition is clearly equivalent to (ii), and so we establish
the equivalence of (i) and (ii).

(ii)	=> (iii). Suppose that (ii) holds. Pick E e 8p arbitrarily. Since E C suppu^
by Lemma 4.3, it follows that Up(x) > 8 for //-almost all x E E, and so, we have

/лр-1(Е) = / Updp > 8p(E).

Je

This is (iii) with c = 8.

(iii)	=> (ii). Suppose that (iii) holds. Take 8 — c (the constant described in
(iii)) and assume that p({x E supply : Up(x) < 5}) > 0 . Then there is a constant
8q (0<8q<8) such that p({x E supp Up : Up(x) < <$o}) > 0. Since p is а-finite, we
find a set E E ЯЛ such that E С {ж G supp Up : «^(x) < ^o} and 0 < p(E) < oo.
Then we see from Lemma 4.3 that E E 8p, and so

cp(E) <vtp~* 1 (ii) (iii) (E) = I Up dp < 8q p(E) < 8 p(E) = c p(E),
Je

which is a contradiction. This shows that p({x E supp Up : Up(x) < £}) = 0 . In
other words, Up(x) > 8 for //-almost all x E supp Up.	□

Case II p> q

Theorem 4.4. Suppose 1 < q < p < oo, and let Cp be a composition operator
from LP(X) into Lq(Y). Then the following assertions are equivalent:

(i) (7^ has closed range.

(ii) Cp has finite rank.

(iii) Up(x) = 0 for p-almost all x E B, and the set {n E N : иДАп) / 0} is
finite.

(vi) i/(/?-1(B) = 0, and the set {n E N : /лр-1(Ап) 7= 0} is finite.

Proof, (i) о (iii). This equivalence is immediately obtained by combining
Lemma 4.1 and Theorem 2.4.

(iii) <=> (vi). As seen in the proof of Theorem 3.8 ((ii)	(iii)), Up(x) = 0

for //-almost all x E В if and only if i/<p-1(B) = 0. Also, from the equation (3.7)
and the fact that p(An) > 0, it follows that {n E N : Up(An) fO} = {neN :
v<p~ 1(An) / 0}. Thus we obtain the equivalence of (iii) and (vi).

(vi) => (ii). Suppose (vi) holds. Then we easily see that the range C'¥J(LP(X)) is
contained in the subspace spanned by the characteristic functions {xri(4n)}ng^>
where Np = {n E N : <p-1(An) / 0}. Since Np is finite, Cp(Lp(Xf) is finite
dimensional.

(ii) => (i). Trivial.	□
This theorem, as well as Theorem 3.8, has the corollary:

Corollary. Suppose 1 < q < p < oo. IfXis nonatomic and v(Y) > 0, then
there is no composition operator from LP(X) into Lq(Y) that has closed range.

Case III p < q

Theorem 4.5. Suppose 1 < p < q < oo, and let Cv be a composition operator
from LP(X) into Lq(Y). Then the following assertions are equivalent:

(i)	Cv has closed range.

(ii)	has finite rank.

(iii)	The set {nEN: u^lAn) 0} is finite.

(vi) The set {n e N : i/(/?-1(An)	0} is finite.

Proof. From Theorem 3.8, we know that и^(х) — 0 for //-almost all ж € В
and that utp~* 1 (ii) (iii)(B) — 0. Hence our assertions just coincide with those of Theorem
4.4, and their equivalences can be proved in the same way. The only difference is
the use of Theorem 2.5 in place of Theorem 2.4.	□

5. Remarks on the L°° case

So far, we have investigated the multiplication operators from LP(X) into
Lq(X} and the composition operators from LP(X) into £<г(У), excluding the case
that p = oo or q = oo. Here we describe the results in that case. Some of these
results are proven in ways similar to the preceding discussions, and the others may
be obtained quite easily.

Case I p = q = oo

Theorem 5.1.	Suppose и 6 L(X). Then и induces a multiplication operator
Mu from L°°(X) into if and only if и e LOO(X). In this case, Mu is a
bounded linear operator from L°°(X) into L°°(X), and its norm is given by \\MU || =
||u||Loo(X). Also, for a multiplication operator Mu from L°°{X) into L°°{X}, Mu
has closed range if and only if there exists a constant 6 > 0 such that Щя)| > 6 for
p-almost all x e supp u.

Theorem 5.2.	For any nonsingular measurable transformation <p from Y into
X, <p induces a composition operator Cv from L^^X) into LOO(Y'). Then C^
is a bounded linear operator from L°°(X) into L°°(Y) with closed range, and if
p(F)>0, ||C„|| = 1.

Case II p — oo and 1 < q < oo

Theorem 5.3. Suppose 1 < q < oo and и e L(X). Then и induces a multi-
plication operator Mu from L^^X) into Lq(X) if and only if и € Lq(X). In this
case, Mu is a bounded linear operator from L°°(X) into Lq(X), and its norm is
given by ||MU|| — ||u||Lg(X). Also, for any multiplication operator Mu from L°°(X)
into Lq(X), the following assertions are equivalent:

(i) Mu has closed range.

(ii) Mu has finite rank.

(iii) u(x) = 0 for p-almost all x E B, and the set {nEN: u(An) 0} is finite.

Theorem 5.4. Suppose 1 < q < oo and let </? be a nonsingular measurable
transformation from Y into X. Ifv(Y) — oo, then <p never induces a composition
operator from L°°(X) into Lq(Y). On the other hand, ifv(Y) < °°> </ien

induces a composition operator Cp from L°°(X) into Lq(Y). In this case, Cp
is a bounded linear operator from L^tX) into Lq(Y) and its norm is given by
ЦС^Ц = i/(Y)«. Also, for any composition operator Cp from L°°(X) into Lq(Y),
the following assertions are equivalent:

(i)	Cp has closed range.

(ii)	Cp has finite rank.

(iii)	i/<p-1(B) = 0, and the set {n G N : i/<p-1(An) / 0} is finite.

Case III 1 < p < oo and q = oo

Theorem 5.5. Suppose 1 < p < oo and и e L(X\ Then и induces a multipli-
cation operator Mu from LP{X) into L°°(X) if and only if и satisfies the following
two conditions:

(5.1) u(x) = 0 for p-almost all x € B.

|u(An)|p

(5.2)	sup —v -	< oo . In this case, Mu is a bounded linear operator from

n(=N P\An)

LP(X) into L°°(X), and its norm is given by ||MU|| = sup . Also, for any
neN p(An)₽

multiplication operator Mu from LP(X) into L°°(X), the following assertions are
equivalent:

(i) Mu has closed range.

(ii) Mu has finite rank.

(iii) The set {neN: u(An) / 0} is finite.

Theorem 5.6. Suppose 1 < p < oo and let <p be a nonsingular measurable
transformation from Y into X. Then <p induces a composition operator Cp
from LP(X) into L°°(Y) if and only if f(p~x{B) = 0 and inf p(An) > 0, where

neN^

Np = {n € N : u<p~* 1 2 3 4 5(An) 7!= 0}. In this case, Cp is a bounded linear operator

from LP(X) into L°°(Y) and its norm is given by ||C'(1O||P — sup — т. Also, for
p{An)

any composition operator Cp from LP(X) into L°°(Y), the following assertions are
equivalent:

(i) Cp has closed range.

(ii) Cp has finite rank.

(iii) The set {neN: и<р~1(Ап)	0} is finite.

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Department of Mathematical Sciences, Faculty of Science, Shinshu University,
Matsumoto 390, Japan

E-mail address: htakagifflripms.shinshu-u.ac.jp