Composition and multiplication operators between Orlicz function spaces

(1)

R E S E A R C H

Open Access

Composition and multiplication operators

between Orlicz function spaces

Tadeusz Chawziuk

1

_{, Yunan Cui}

2*

_{, Yousef Estaremi}

3

_{, Henryk Hudzik}

1

_{and Radosław Kaczmarek}

1

*_{Correspondence:}

[email protected]

2_{Department of Mathematics,}

Harbin University of Science and Technology, Harbin, 150080, P.R. China

Full list of author information is available at the end of the article

Abstract

Composition operators and multiplication operators between two Orlicz function spaces are investigated. First, necessary and suﬃcient conditions for their continuity are presented in several forms. It is shown that, in general, the Radon-Nikodým derivatived(μ_d◦_μτ–1)(s) need not belong toL∞() to guarantee the continuity of the composition operatorcτx(t) =x(τ(t)) fromL() intoL(). Next, the problem of

compactness of these operators is considered. We apply a compactness criterion in Orlicz spaces which involves compactness with respect to the topology of local convergence in measure and equi-absolute continuity in norm of all the elements of the set under consideration. In connection with this, we state some suﬃcient conditions for equi-absolute continuity of the composition operatorcτand the

multiplication operatorMwfrom one Orlicz space into another. Also the problem of necessary conditions is discussed.

MSC: Primary 47B33; secondary 46E30

Keywords: composition operator; multiplication operator; Orlicz spaces

1 Introduction

Since the early s composition operators have been a subject of study of many mathe-maticians or physicists. At the beginning they were used to solve problems in mathemat-ical physics and classmathemat-ical mechanics [, ], or to study ergodic transformations [, ]. Up until now many Ph.D. theses have been defended (for instance: Boyd [], Gupta [], Ridge [], Schwartz [], Singh [], Swanton [], Veluchamy []), numerous books published [, ], and innumerable papers printed on the composition operator or the weighted composition operator in various function spaces,e.g.,Lp_{spaces ([–], and others),}

Or-licz spaces ([–], and others), Musielak-OrOr-licz spaces [, ], Musielak-OrOr-licz spaces of Bochner type ([] or []), Orlicz-Lorentz spaces [–], Hilbert spaces [–] and many other types of spaces (for instance: [–]). The multiplication operator has also been a subject of research of many mathematicians (see for instance: [–]). For more details as regards the historical background we refer the reader to [].

Let (,,μ) be a non-atomic,σ-ﬁnite and complete measure space and letτ:→

be a measurable function,i.e., a mapping such thatτ–₍_A₎_∈_{if and only if}_A_∈_{for any}

A⊂, whereτ–(A) is the counterimage ofA. In the whole paper we will assume thatτ

is non-singular, that is,μ(τ–₍_A_{)) =  provided}_μ₍_A_{) = . The last assumption guarantees}

(2)

that the measureμ◦τ–_{deﬁned for any}_A_∈_{by the formula}

μ◦τ–(A) =μτ–(A)

is absolutely continuous with respect to the measure μ (what is usually denoted by

μ◦τ–_μ_{). Then the Radon-Nikodým theorem implies the existence of a non-negative}

locally integrable functionhτonsuch thatμ◦τ–(A) =

Ahτ(s)dμ(s) for anyA∈. The

functionhτ(s) =d(μ◦τ

–₎

dμ (s) is called the Radon-Nikodým derivative of the measureμ◦τ

–

with respect to the measureμ. Let us remark that even if the Radon-Nikodým derivative is unbounded, the corresponding composition operator acting from an Orlicz space onto itself can still be continuous (see []).

LetL₍_{) =}_L₍_,_,_μ_{) be the space of all (abstract classes of )}_{-measurable functions}

fromintoRwith respect to the equivalence relation:x∼yif and only ifx(t) =y(t) for

μ-a.e.t∈.

With the help of a non-singular and measurable mappingτ:→one deﬁnes onL

the composition operator

(cτx)(t) :=x

τ(t)

for anyt∈and anyx∈L().

Letw∈L(,,μ) be a strictly positive function. We deﬁne the multiplication operator

Mw:L→Lby the formula

(Mwx)(t) :=w(t)x(t)

for anyt∈and anyx∈L₍_).

It is obvious thatcτx∈L() andMwx∈L() ifx∈L().

Remark . We do not assume, if not speciﬁcally stated otherwise, that the mappingτis a surjection,i.e.,τ() =.

A function:R→R+= [,∞) is said to be an Orlicz function ifis convex, even,

continuous, vanishing only at .

The complementary function in the sense of Young to an Orlicz functionis deﬁned to be the function∗: [,∞)→[,∞] such that∗(u) =sup_v>{uv–(v)}.

A functionϕfrom×RintoR+ such thatϕ(t,·) is an Orlicz function forμ-a.e.t∈

andϕ(·,u) is a-measurable function for everyu∈Ris called a generalized Orlicz function or a Musielak-Orlicz function. The Musielak-Orlicz spaceLϕ₌_Lϕ₍_,_,_μ_{) is the}

space of all (equivalence classes of )-measurable functionsx:→Rsuch that

Iϕ(λx) =

ϕt,λx(t)dμ<∞

for someλ>  depending onx. The Musielak-Orlicz space equipped with the Luxemburg norm

xϕ=inf

λ>  :Iϕ

x

λ

(3)

is a Banach space (cf.[, ], and in the case of Orlicz spaces also [–]). It is obvious that Orlicz functions are Musielak-Orlicz functions, and consequently, Orlicz spaces are Musielak-Orlicz spaces. For instance, an Orlicz weighted spaceL

h() is the

Musielak-Orlicz spaceLϕ₍_{) generated by}_ϕ₍_t_,_u_{) =}₍_u₎_h₍_t_{) for any}_t_∈_{and any}_u_∈_R.

Throughout the paper, we will make use of Ishii’s theorem from [].

Theorem . For Musielak-Orlicz spaces Lϕ₍₎_{and L}ψ₍₎_{over a non-atomic measure} space= (,,μ),the inclusion Lϕ₍₎_⊂_Lψ₍₎_{holds if and only if there exist k}_,_K_{> } and c(·)∈L₍₎_{such that}

ψ(t,kξ)≤Kϕ(t,ξ) +c(t)

for allξ≥and a.e.t∈.

For anyA∈, byI(x,A) we mean the value of the modularIatxin the Orlicz space L₍_A_{) generated by the Orlicz function}_{over the measure space (}_A_,_∩_A_,_μ_|

A). In the

case whenA=, we will writeI(x) instead ofI(x,).

2 Continuity of the composition operatorcτfromLintoLand fromL ontoL

We are interested in ﬁnding necessary and suﬃcient conditions for the continuity of the composition operatorcτ from the Orlicz spaceL() =L(,,μ) equipped with the

Luxemburg norm into the Orlicz space L₍_{) =}_L₍_,_,_μ_{) endowed with the}

corre-sponding Luxemburg norm. The following fact will be very helpful.

Fact . For an arbitrary function x∈L₍_{), we have} _c

τx∈L() if and only if x∈ L

h(τ()), whereLh(τ()) =Lh(τ(),∩τ(),μ|∩τ()) is a weighted Orlicz space with

the weight functionh(s) =dμ◦τ_d_μ–(s).

Proof For anyx∈L(), we have

I(cτx,) =

cτx(t)

dμ(t)

=

xτ(t)dμ(t)

=

τ()

x(s)dμ◦τ–(s)

=

τ()

x(s)h(s)dμ(s) =I,h

x,τ().

Theorem . If the quadruple,,h,τsatisﬁes the condition ∃

K>A∈∩τ∃ ()

μ(A)=

∃

g∈L+(τ())s∈τ(∀)\Au∀≥(u)h(s)≤(Ku) +g(s), ()

or,equivalently,L₍_τ₍₎₎_⊂_L

(4)

Moreover,ifμ(\τ()) = ,then condition()is necessary for the continuity of the com-position operator cτ from L()into L().

Proof We will show that if condition () is satisﬁed thencτx∈L() wheneverx∈L(), i.e.,x∈L

h(τ()) wheneverx∈L

₍_{) (see Fact .).}

Assume thatx∈L₍_{). Applying condition () and Fact ., we obtain}

I

cτx

K( +gL₍_τ₍₎₎)x

,

≤ 

 +gL₍_τ₍₎₎

I

cτx Kx,

= 

 +gL₍_τ₍₎₎I,h

x

Kx,τ()

≤ I(

x

x,τ()) +gL(τ())  +gL₍_τ₍₎₎

≤ 

 +gL₍_τ₍₎₎

 +gL₍_τ₍₎₎= ,

which shows thatcτ acts fromL() intoL() and cτx≤K

 +gL₍_τ₍₎₎x,

which ﬁnishes the proof of the ﬁrst part of the theorem.

Now assume thatμ(\τ()) = . If the inclusion in the assumption of the theorem fails to hold, then there exists a functionxbelonging toL₍_τ₍_{)) =}_L₍_{) but not belonging}

toL

h(τ()). In virtue of Fact ., we obtainx∈L

₍_{) and, simultaneously,}_c

τx∈/L(),

hencecτ does not even act fromL() intoL().

The preceding theorem can be formulated in a diﬀerent language, which in some situa-tions might be more useful.

Theorem . The composition operator cτ :L()→L()is continuous if the following simple condition is satisﬁed:

τ()

χh(s)dμ(s) <∞, ()

where h=dμ_d◦_μτ– andχis the function complementary in the sense of Young to the function

◦K–,with K being the constant from condition()of Theorem..Ifμ(\τ()) = ,

then condition()is necessary for the continuity of cτ.

Proof It is not too diﬃcult to see that,, andhsatisfy condition () if and only if

sup

u≥

(u)h(s) –(Ku)∈L₊τ().

But

sup

u≥

(u)h(s) –(Ku)=sup

v≥

vh(s) –K–(v)

(5)

Remark . Notice that ifb(χ) :=sup{u≥ :χ(u) <∞}<∞then condition () implies thathL∞(τ())≤b(χ) <∞, that is, the Radon-Nikodým derivativeh=dμ◦τ

–

dμis es-sentially bounded. Moreover, it is easy to see that the integral () can be ﬁnite for some

h∈/L∞(τ()) if and only if the functionχhas only ﬁnite values (for the case when=

see []; the proof in the case when=is similar).

The next theorem states a necessary and suﬃcient condition in order thatχis such a function.

Theorem . The functionχ= (◦K–₎∗_,_{with K}_{> ,}_{assumes only ﬁnite values}₍_i.e._,

b(χ) =∞)if and only iflim inft→∞(Kt)(t) =∞.

Proof Suﬃciency. Let lim inft→∞(Kt)(t) = ∞. Then limt→∞

(Kt)

(t) = ∞ and so

lim_t_→∞(K–(t))

t =∞. Take an arbitraryv> . In virtue of the last equality, there exists

uv>  such that for allu≥uvthe inequality

(K–(u))

u ≥v

holds. Hence

χ(v) :=sup

u≥

uv–K–(u)=sup

u≥

u

v–(K

–₍_u₎₎

u

= sup

u∈[,uv]

uv–K–(u)<∞

as a supremum of a continuous function on a compact interval.

Necessity. Assume thatlim inft→∞(Kt)(t) <∞forK> . Thenlim inft→∞

(K–(t))

t <∞

forK> , and so there existc>  and a sequence of positive numbers (un)∞n= such that

un→ ∞asn→ ∞andlimn→∞(K

–_(u n))

un =c. Consequently,c

_∈₍_c_,_∞_{) can be found}

such that forn∈Nlarge enough we have

K–(un)

≤cun.

Therefore, takingv>c, we get

unv–

K–(un)

≥unv–cun=un

v–c

forn∈Nlarge enough, whence

sup

u≥

uv–K–(u)≥sup

n∈N

unv–

K–(un)

≥v–csup

n∈Nun=∞,

which ﬁnishes the proof of the theorem.

Now we show that if the functionχfrom Theorem . assumes only ﬁnite values,i.e.,

b(χ) =∞, then it may happen that the composition operatorcτ fromL() intoL() is

(6)

Example . Let= (, ], (,,μ) be the Lebesgue measure space andAn= (_n+ ,_n] for

everyn∈N. Then∞_n=An=. Letχbe the function from Theorem ., letb(χ) =∞

andun>  be such thatχ(un) =

√

nfor everyn∈N. Sinceun ∞asn→ ∞, there exists

m∈Nsuch thatun≥ for anyn>m. Deﬁneτ: (, ]→(, ] by the formula

τ(t) =

⎧ ⎨ ⎩

t, fort∈Anwithn≤m, t

un, fort∈An, wheren>m.

Let us denotevn=  ifn≤mandvn=unifn>m. Thenτ–:τ()→is deﬁned by the

formulaτ–₍_s_{) =}_v

nsfor anys∈τ(An). Consequently, for anyA∈∩τ(), we have

μ◦τ–(A) :=μτ–(A)=μ

_∞

n=

τ–A∩τ(An)

=

∞

n=

μτ–A∩τ(An)

= ∞ n= μ vn A∩ 

vn

(An)

=

∞

n=

μ(vnA∩An). ()

Let us deﬁne

h(s) =

∞

n=

vn1

unAn(s). ()

Then, for anyA∈∩τ(),

A

h(s)dμ(s) =

_∞ n=A∩τ(An)

h(s)dμ(s) =

∞

n=

A∩ vn(An)

vndμ(s)

=

∞

n=

vnμ

A∩ 

vn An = ∞ n=

μ(vnA∩An). ()

By inequalities () and (), we get

μ◦τ–(A) =

A

h(s)dμ(s), ∀A∈∩τ(),

which means thath=d(μ◦τ_d_μ–). By formula () and the deﬁnition ofvn, we have

τ()

χh(s)=

∞

n=

χ(vn)μ

 vn An = m n=

μ(An) +

(7)

≤ +

∞

n=m+

√ nμ(An)

≤ +

∞

n=m+

√ n n(n+ )

<  +

∞

n=



n/ <∞.

In virtue of Theorem ., we conclude that the composition operatorcτ fromL() into L₍_{) is continuous. However, since}_u

n→ ∞asn→ ∞, we havehL∞(τ())=∞.

Theorem . The composition operator cτ acts continuously from L(,,μ) onto L₍_,_,_μ₎_{if and only if}_μ₍_\_τ₍_{)) = }_{and the following two conditions are jointly} sat-isﬁed:

(i) ∃K>∃A∈∩

μ(A)= ∃g∈L 

+()∀s∈\A∀u≥(u)h(s)≤(Ku) +g(s); (ii) ∃K>∃A∈∩

μ(A)= ∃p∈L 

+()∀s∈\A∀u≥(u)≤(Ku)h(s) +p(s),

where h(s) = dμ◦τ_d_μ–(s)forμ-a.e.s∈.

Proof Obviously, the conditionμ(\τ()) =  is necessary forcτ(L()) =L(), so in

the further part of the proof we assume this condition holds. Therefore, by Fact ., we know thatcτ acts fromL() ontoL() if and only ifL() =L_h(). Equivalently, this

holds if and only if we have two inclusions:L₍₎_⊂_L

h() andL

h()⊂L

₍_{). The ﬁrst}

inclusion holds if and only if condition (i) is satisfied and the reverse inclusion holds if and only if condition (ii) is satisfied (see [] and []), and this finishes the proof.

It is interesting and proﬁtable to observe that the preceding theorem can be written in the following form.

Theorem . The composition operator cτ acts continuously from L()onto L()if and only ifμ(\τ()) = and for some K> the following two conditions are jointly sat-isﬁed:

() χ(h(s))dμ(s) <∞; () h(s)q(_h(s) )dμ(s) <∞,

whereχis the function complementary in the sense of Young to the function◦K–_,_{q is}

the function complementary in the sense of Young to the function ◦K–_,_{and h}₍_s_{) =} dμ◦τ–

dμ (s)forμ-a.e.s∈.

Proof In the proof of Theorem . we already showed that condition (i) from Theorem . is equivalent to condition (). So, the proof will be ﬁnished if we show that condition () is equivalent to condition (ii) from Theorem ..

It is easy to see that condition (ii) is equivalent to the fact thatq∈L

+(), where

q(s) =sup

u≥

(8)

for alls∈. Sinceh(s)=  forμ-a.e.s∈, we have

q(s) =h(s)sup

u≥

(u)

h(s) –(Ku)

=h(s)sup

v≥

v h(s)–

K–(v)

=h(s)◦K–∗



h(s)

forμ-a.e.s∈, where (◦K–)∗ is the complementary function to the function◦

K–, which ﬁnishes the proof.

3 Continuity of the multiplication operatorMwfromLintoLand fromL ontoL

We will state criteria in order thatMw mapL(,,μ) intoL(,,μ), whereand

are distinct Orlicz functions. Note thatMwx∈L(,,μ) means that there isλ> 

such thatI(λw(t)x(t)) =

(λw(t)x(t))dμ(t) <∞. This is equivalent to the fact thatx∈ Lw₍_,_,_μ_{), where}_Lw₍_,_,_μ_{) is a Orlicz space generated by the} Musielak-Orlicz functionw(t,u) :=(w(t)u). Let us begin with the following.

Theorem . The multiplication operator Mwmaps L(,,μ)into L(,,μ)if and

only ifχK(t, )dμ(t) <∞for some K> ,whereχK(t,u)is,for ﬁxed t∈,the function

complementary in the sense of Young to the function◦_w(t)K –with respect to u.

Proof The fact thatMw:L(,,μ)→L(,,μ) means thatL(,,μ)⊂Lw(,

,μ), which, by Theorem ., holds if and only if there areA∈withμ(A) = , a constant

K> , and a functionh∈L

+(,,μ) such that the inequality

w(t)u≤(Ku) +h(t) () holds for allt∈\Aand allu∈R. It is easy to see that this is equivalent to the fact that

for someK>  the functionh˜Kdeﬁned by the formula

˜

hK(t) =sup u∈R

w(t)u–(Ku)=sup

u≥

w(t)u–(Ku) ()

is integrable over. In fact,h˜Kis the smallest function such that condition () holds with

˜

hKin place ofh. Setting in ()u= –_(v)

w(t) , we get

˜

hK(t) =sup v>

v–

K w(t)

–₍_v₎

=χK(t, ),

whereχK(t,u) is the function complementary in the sense of Young to◦_w(t)K –.

There-fore, the fact thatMwmaps continuouslyL(,,μ) intoL(,,μ) is equivalent to the

fact thatχK(t, )dμ(t) <∞.

Theorem . The multiplication operator Mw maps L(,,μ) onto the whole of

(9)

(i) χK(t, )dμ(t) <∞for someK> ,whereχK(t,u)is the function deﬁned in Theorem.;

(ii) UK(t, )dμ(t) <∞for someK> ,whereUK(t,u)is,for ﬁxedt∈,the function complementary in the sense of Young,with respect tou,to the function

(◦Kw(·)–)(t,u) =(Kw(t)–(u)).

Proof It is obvious thatMwmapsL(,,μ) ontoL(,,μ) if and only ifμ(\suppw) =

 and {w(t)x(t) : x∈ L_} ₌_L₍_,_,_μ_{). The space on the left-hand side of the last}

equality is the Musielak-Orlicz space Lw₍_{), where}

w(t,u) =(_w(t)u ). Observe that

Lw₍_,_,_μ_{) =}_L₍_,_,_μ_{) is equivalent to}_L₍_,_,_μ_{) =}_Lw₍_,_,_μ_{), where} w(t,u) =

(w(t)u). Therefore, the assumption thatMwmapsL(,,μ) ontoL(,,μ) means

that L₍_,_,_μ_{) =}_Lw₍_,_,_μ_{), that is, the inclusions} _L₍₎_⊂_Lw₍_{) and} _Lw₍₎⊂

L₍_{) hold. The preceding theorem established that the ﬁrst inclusion is equivalent to}

condition (i). We need only to prove that the reverse inclusion is equivalent to (ii). How-ever, by Ishii’s theorem, the inclusionLw₍₎⊂_L₍_{) holds if and only if there is a} con-stantK> , a setA∈withμ(A) = , andg∈L

+(,,μ), such that

(u)≤Kw(t)u+g(t)

for allt∈\Aandu≥. The last condition is equivalent to the condition

sup

u≥

(u) –Kw(t)u∈L₊(,,μ).

But note that

sup

u≥

(u) –Kw(t)u=sup

v≥

v–Kw(t)–(v) =UK(t, ),

whereUK(t,u) is, for ﬁxedt∈, the function complementary in the sense of Young, with

respect tou, to the functionM(t,u) =(Kw(t)–₍_u_{)). This ﬁnishes the proof.}

4 Compactness of the composition operatorcτfrom one Orlicz space into another

We begin with some notions that will be useful in the following. Let (,,μ) be a non-atomic, complete andσ-ﬁnite measure space. We say that functions in a setAcontained in the Musielak-Orlicz spaceL₍_{) have equi-absolutely continuous norms if for any real}

numberε>  there exist a setBε∈withμ(Bε) <∞and a real numberδ=δ(ε) >  such

that for any functionx∈Awe havexχ_\Bε<εandxχB<εwheneverB∈∩Bε

andμ(B) <δ.

LetL₍_{) =}_L₍_,_,_μ_{) and}_L₍_{) =}_L₍_,_,_μ_{) be distinct Orlicz spaces. We say that}

the operatorT:L₍₎_→_L₍_{) is equi-absolutely continuous if for any bounded set}_A_⊂ L₍_{) all functions of the set}_T₍_A₎_⊂_L₍_{) have equi-absolutely continuous norms.}

(10)

Theorem .(Theorem . in []) Let(,,μ)be a non-atomicσ-ﬁnite measure space and letϕbe a Musielak-Orlicz function.If the functions in a set A⊂Lϕ₍₎_{all have} equi-absolutely continuous norms and A is relatively compact with respect to local convergence in measure,then A is relatively compact in Eϕ₍_),_{the subspace of absolutely continuous} functions in Lϕ₍_).

Conversely,if a set A⊂Eϕ₍₎_{is relatively compact}_,_{then all the functions in A have} equi-absolutely continuous norms and A is relatively compact with respect to local convergence in measure.

In the proof of the forthcoming theorem we will need the following.

Lemma .(Lemma . in []) Let the measureμbe atomless and let a sequence{αi}of

positive numbers and a sequence{ai}of measurable,ﬁnite,non-negative functions inbe

given,satisfying the inequalities

ai(t)dμ≥iαi for i= , , . . . .

Then there exist an increasing sequence {ik} of positive integers and a sequence{Ak} of

pairwise disjoint sets fromsuch that

Ak

aik(t)dμ=αik for k= ,  . . . .

The following theorem will be of great importance in proving necessary and sufficient conditions for the compactness of the composition operatorcτ :L()→L(). Theorem . Let(,,μ)be a finite or infinite butσ-finite non-atomic and complete measure space andτbe such thatμ(τ–(A)) <∞wheneverμ(A) <∞for any A∈∩τ().

Then the composition operator cτ:L()→L()is equi-absolutely continuous whenever the following condition is satisﬁed:

∀

σ>A∈∩τ∃ ()

μ(A)=

∃

gσ∈L+(τ())

∀

s∈τ()\Au∀≥(u)h(s)≤(σu) +gσ(s). ()

Condition()is necessary for the equi-absolute continuity of cτ ifμ() <∞.

Proof Suﬃciency. First we prove that for anyε>  there exists a setD∈withμ(\D) <

∞such that all the functions in the set{cτx:x∈S(L)}, whereS(L) is the unit sphere

ofL_{, satisfy the condition}₍_c

τx)χD<ε.

Letσ>  be a number such that ( +σ)σ<εand letgσ be a function from condition

() corresponding toσ. Sincegσ ∈L+(τ()), there exists a setC∈∩τ() such that

gσχC<σandμ(τ()\C) <∞. DeﬁningD=τ–(C), we have for any functionx∈S(L)

I

cτx

σ χD

=

cτx(t)

σ χD(t)

dμ(t)

=

D

x(τ(t))

σ

(11)

=

τ(D)

x(s)

σ

dμ◦τ–(s)

=

C

x(s)

σ

h(s)dμ(s)

≤

C

x(s)dμ(s) +

C

gσ(s)dμ(s)

≤ +σ. ()

By () we get

I

(cτx)χD

( +σ)σ

≤ 

 +σI

cτx

σ χD

≤ 

 +σ( +σ) = 

for anyx∈S(L_{). Consequently,}₍_c

τx)χD≤( +σ)σ<εfor anyx∈S(L), which

ﬁn-ishes the ﬁrst part of the proof.

Now we show the following implication:

∀

σ>δ=∃δ(ε)B⊂τ∀()\Dx∈S(L∀₎μ(B) <δ ⇒

(cτx)χB<ε,

whereDis the set from the ﬁrst part of this proof. Take anyx∈S(L_{). Since}_g

σ ∈L₊(τ()), it is obvious that there isδ=δ(σ) >  such that,

ifC∈∩(τ()\D) andμ(C) <δ, then_Cgσ(s)dμ(s) <σ. LetB=τ–(C). Then applying

condition (), we get

I

cτx

σ χB

= cτx(t)

σ χB(t)

dμ(t)

=

B

x(τ(t))

σ

dμ(t)

=

τ(B)

x(s)

σ

dμ◦τ–(s)

=

C

x(s)

σ

h(s)dμ(s)

≤

C

x(s)dμ(s) +

C

gσ(s)dμ(s)

≤ +σ. ()

Now, by convexity of the modularIand the fact thatIvanishes at zero we haveI(λx)≤

λI(x) for anyx∈L() andλ∈[, ]. From this fact and from (), we get

I

(cτx)χB

( +σ)σ

≤ 

 +σI

cτx

σ χB

≤ 

 +σ( +σ) = .

Hence(cτx)χB≤( +σ)σ<ε. Sinceσdepends only onε,δ=δ(σ) depends only onε,

and so the proof of suﬃciency is ﬁnished.

Necessity. Assume thatμ() <∞and for anyσ>  deﬁne the function

hσ(s) =sup

u≥

(12)

hσ(s) is a non-negative (since foru= , we have()h(s) –(σ) = ) measurable

func-tion.

Suppose condition () is not satisﬁed. Then there isσ>  such that

τ()

hσ(s)dμ(s) = +∞.

Let{ri}∞i=be a sequence of all non-negative rational numbers withr= . By the continuity

of the Orlicz functionsand, we have

hσ(s) = sup i=,,...

(ri)h(s) –(σri)

.

Let us write

hσ,n(s) = max ≤i≤n

(ri)h(s) –(σri)

.

It is obvious thathσ,n≥ andhσ,nare measurable functions such that hσ,nhσ as

n→+∞μ-a.e. inτ(). By Beppo Levi’s theorem, we have

τ()

hσ,n(s)dμ(s)

τ()

hσ(s)dμ(s).

Hence there exists a subsequence{hσ,nk} ⊂ {hσ,n}satisfying

τ()

hσ,nk(s)dμ(s)≥ k_.

Without loss of generality we may assume that_τ₍₎hσ,n(s)dμ(s)≥nfor eachn∈N∪

{}. It is clear that for eachs∈τ() and eachn∈N∪{}there existsrn(s)∈ {r,r,r, . . . ,rn}

such that

hσ,n(s) =

rn(s)

h(s) –σrn(s)

.

Hence

τ()

rn(s)

h(s)dμ(s) =

τ()

hσo,n(s)dμ(s) +

τ()

σrn(s)

dμ(s)≥n_.

Applying Lemma ., we conclude that there is a sequence of sets{k} ⊂τ() withi∩

j=∅fori=jsuch that

n

rn(s)

h(s)dμ(s) = .

(13)

Since, by assumption, μ() <∞, we have ∞_n=μ(n) =μ(

∞

n=n)≤μ() <∞,

whencelimn→∞μ(n) = . This means that{¯rn}+n=∞does not have equi-absolutely

contin-uous norms inE

h(). Yet, using the fact thatgσ,n≥ (n∈N∪ {}), we get the following:

r¯n(s)

h(s)≥σ¯rn(s)

, n∈N∪ {}.

SoI(σ¯rn)≤, that is,¯rn≤_σ_, which means that the operatorcτ :L()→L() is

not equi-absolutely continuous. Hence we proved that condition () is necessary for the

equi-absolutely continuity ofcτ.

From Theorem ., applying Theorem . and the deﬁnition of a compact operator, we directly get the following.

Theorem . If(,,μ)is a non-atomic complete finite or infinite butσ-finite measure space andτ satisfies the assumption from Theorem.,then the composition operator cτ from L₍₎_{into L}₍₎_{is compact whenever the set c}

τ(S(L))is relatively compact with respect to local convergence in measure and condition()from Theorem.is satisﬁed.

Under the assumption thatμ() <∞,if the composition operator cτ from L()into E₍₎_{is compact then the set c}

τ(S(L))is relatively compact with respect to convergence in measure and condition()is satisﬁed.

In the case whenhas inﬁnite measure, we were unable to show that () is a necessary condition for the equi-absolute continuity of the composition operatorcτ. Instead, we can

deduce a slightly diﬀerent (and weaker) condition, as the following theorem states.

Theorem . Assume thatμ() =∞andμ(\τ()) = . If the composition operator cτ :L()→L()is equi-absolutely continuous,then the condition

∀ λ> A∃∈

μ(A)=

∃

Kλ>gλ∈L∃+()

∀

s∈\Au∀≥(λu)h(s)≤Kλ(u) +gλ(s) ()

is satisﬁed.

Proof We can assume without loss of generality thatτ() =. First, notice that if,, andhsatisfy condition () thenL₍₎_⊂_E

h(), where

E_h() =

x∈L() : ∀

λ>I,h(λx) =

λx(t)h(t)dμ(t) <∞

is the subspace of absolutely continuous elements ofL

h(). The inclusion results from the

assumption thatcτis an equi-absolutely continuous operator,i.e., for any bounded setA⊂ L₍_{), the functions of the set}_c

τ(A)⊂L() all have equi-absolutely continuous norms,

and the observation that, for anyx∈L₍_{), the singleton set}_{_x_}_{is bounded, and thus}_x

has an equi-absolutely continuous norm, which means thatxis an absolutely continuous element ofL

h(),i.e.,x∈E

h().

Further, observe that

L() =

λ>

Lλ,∗₍_), _E

h() =

λ>

Lλ,∗

(14)

where for any λ>  we deﬁne λ(u) =(u) and L,∗() ={x∈L() :I(x) <∞} is a

Musielak-Orlicz class (λ and L_h,∗() are deﬁned analogously). Hence the inclusion L₍₎_⊂_E

h() can be expressed as

λ>

Lλ,∗₍₎⊂

λ>

Lλ,∗

h ().

This means that, for anyλ> , the Musielak-Orlicz classLλ,∗() is contained in all of

Lλ,∗

h () (λ> ). In particular, takingλ = , we see thatL,∗() is contained in all the

Musielak-Orlicz classesLλ,∗

h () (λ> ). By Theorem . in [], this is equivalent to

∀ λ> A∃∈

μ(A)=

∃

Kλ>gλ∈L∃+()

∀

s∈\Au∀≥(λu)h(s)≤Kλ(u) +gλ(s),

which ﬁnishes the proof.

From the preceding theorem, applying Theorem ., we can deduce the following nec-essary condition for the compactness of the composition operatorcτ:

Theorem . Assume thatμ() =∞ andμ(\τ()) = .If the composition operator cτ :L()→E()is compact then the following conditions are jointly satisﬁed:

() the setcτ(S(L))is relatively compact with respect to local convergence in measure;

() the functionsandsatisfy condition().

Remark . If there existsε>  such that the set

Bε:=

t∈τ() : ∀

u≥(u)h(t) >(εu)

has positive measure, then no composition operator cτ :L()→L() over a

non-atomic measure space (,,μ) is compact.

Proof of Remark. Assume that there existsε>  such that the measure of the setBεis

positive. Then in the setBεwe can ﬁnd a sequence of measurable and pairwise disjoint

sets{Bn}inτ() having positive and ﬁnite measure. Deﬁne

xn=

χBn

χBn =–



μ(Bn)

χBn.

Obviously,I(xn) = , soxn=  for alln∈N. We have

I

εxn

xn◦τ

,

=

τ()

εxn(t)

xn◦τ

dμ(t)

<

τ()

xn(t)

xn◦τ

h(t)dμ(t)

=

(xn◦τ)(t)

xn◦τ

dμ(t) =I

cτxn

,

(15)

Therefore,

εxn

xn◦τ

≤,

i.e.,cτxn≥εxn=εbecausexn= . Since the supports ofxnare pairwise

dis-joint, for alln,m∈N,m=nwe get

cτxn–cτxm=cτ(xn–xm)≥max

cτxn,cτxm

≥ε.

Hence the sequence{cτxn}∞n=has no Cauchy subsequence, that is,cτ(S(L())) is not

rel-atively compact. Consequently, no composition operatorcτ fromL() intoL() over

a non-atomic measure space (,,μ) is compact.

5 Compactness of the multiplication operatorMwfrom one Orlicz space into another

We state a suﬃcient condition for the compactness of the multiplication operatorMw:

L₍₎_→_L₍_).

Theorem . Let(,,μ)be a non-atomic complete,finite or infinite butσ-finite measure space and let,be two Orlicz functions and w∈L

+(,,μ).If the triple,,w satisﬁes

the condition

∀ σ> Aσ∃∈

μ(Aσ)=

∃

gσ∈L+()

∀

t∈\Au∀≥

w(t)u≤(σu) +gσ(t) ()

then the multiplication operator Mwfrom L()into L()is equi-absolutely continuous.

Proof First we show that givenε>  there exists a set B∈ withμ(B) <∞such that

xχ\B<εfor any functionx∈S(L()).

Takeε>  and letσ >  be such thatσ( +σ) <ε. Letgσ be a function from condition

() corresponding toσ. Sincegσ ∈L₊(), there existsB∈ withμ(B) <∞such that gσχ\BL₍₎<σ. Then

I

Mwx

σ χ\B

≤I(xχ\B) +

\B

gσ(t)dμ(t)≤ +σ,

whence

I

Mwx

σ( +σ)χ\B

≤ 

 +σI

Mwxχ\B

σ

≤ 

 +σ( +σ) = ,

that is,Mwxχ\B≤σ( +σ) <ε.

Next, we show that forε>  there existsδ=δ(ε) >  such that for anyD⊂Band any functionx∈S(L_{), if}_μ₍_D_{) <}_δ_then_M

wxχD<ε.

Letε>  and letσ >  be such thatσ( +σ) <ε. By the absolute continuity ofgσ in L(∩B), there existsδ=δ(σ) such thatgσχCL₍₎<σ wheneverC⊂Bandμ(C) <δ.

Then, by condition (),

I

Mwx

σ χC

(16)

whence

I

Mwx

σ( +σ)χC

≤ 

 +σI

MwxχC

σ

≤,

and soMwx≤σ( +σ) <ε, which ﬁnishes the proof.

Remark . Let us note that in the case when=, Theorem . can only hold when

∈(∞).

Indeed, if Theorem . holds, then the operatorMwacts, in fact, fromL() intoE().

However, if we assume that∈/(∞) andw∈/L∞(,,μ), then deﬁning the set

A=t∈:w(t)≥,

we haveμ(A) > . Therefore, we can buildx∈L₍_{) such that}_supp_x_⊂_A_,_I

(x)≤, and I(λx) =∞for anyλ>  (see [] and []). HenceI(Mwx)≥I(x) =∞, which means

thatMwx∈/E(), soMwdoes not act fromL() intoE().

Applying Theorem ., we directly get from Theorem . and the deﬁnition of a compact operator the following.

Theorem . If(,,μ)is a non-atomic complete finite or infinite butσ-finite measure space,then the multiplication operator cτ form L()into L()is compact whenever the set Mw(S(L))is relatively compact with respect to local convergence in measure and

condition()from Theorem.is satisﬁed.

Theorems . and . resemble closely the suﬃciency part of Theorems . and . for the composition operator. Similarly, we will formulate necessary conditions for the equi-absolute continuity of the multiplication operator: one in the case whenμ() <∞and the other in the case whenμ() =∞. The respective proofs proceed along the lines of the proofs for the composition operator, and therefore will be omitted.

Theorem . If(,,μ)is a ﬁnite non-atomic and complete measure space and the mul-tiplication operator Mw:L()→L()is equi-absolutely continuous then the following

condition is satisﬁed:

∀ σ> A∃∈

μ(A)=

∃

gσ∈L+()

∀

t∈\Au∀≥

w(t)u≤(σu) +gσ(t).

Theorem . If(,,μ)is an inﬁnite but σ-ﬁnite non-atomic and complete measure space and the multiplication operator Mw:L()→L()is equi-absolutely continuous,

then the following condition is satisﬁed:

∀ λ> A∃∈

μ(A)=

∃

Kλ>gλ∈L∃+()

∀

t∈\Au∀≥

w(t)λu≤Kλ(u) +gλ(t).

(17)

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Although the subject of this paper was initiated by HH who was heading and controlling the scientiﬁc research of the other authors of the paper, all authors contributed equally to this paper. All authors read and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics, Faculty of Mathematics and Computer Science, Adam Mickiewicz University in Pozna ´n,}

Pozna ´n, Poland.2_{Department of Mathematics, Harbin University of Science and Technology, Harbin, 150080, P.R. China.} 3_{Department of Mathematics, Payame Noor University (PNU), P.O. Box: 19395-3697, Tehran, Iran.}

Acknowledgements

The second author is supported by NFS of Heilong Jiang province (A2015018).

Received: 11 August 2015 Accepted: 14 January 2016 References

1. Koopman, BO: Hamiltonian systems and transformations in Hilbert space. Proc. Natl. Acad. Sci. USA17, 315-318 (1931)

2. Koopman, BO, Von Neumann, J: Dynamical systems of continuous spectra. Proc. Natl. Acad. Sci. USA18, 255-263 (1932)

3. Halmos, PR: Lectures on Ergodic Theory. Chelsea, New York (1965)

4. Halmos, PR, Von Neumann, J: Operator methods in classical mechanics II. Ann. Math.43, 332-350 (1942)

5. Boyd, DM: Composition operators on the Bergman spaces and analytic function spaces on annulus. Thesis, University of North Carolina (1974)

6. Gupta, DK: Composition operators on weighted sequence spaces. Thesis, University of Jammu (1979) 7. Ridge, WC: Composition operators. Thesis, Indiana University (1969)

8. Schwartz, HJ: Composition operators onHp_{. Thesis, University of Toledo (1969)}

9. Singh, RK: Composition operators. Thesis, University of New Hampshire (1972) 10. Swanton, DW: Composition operators onHP₍_D_{). Thesis, Northwestern University (1974)}

11. Veluchamy, T: Composition operators onLp_{-spaces. Thesis, University of Jammu (1983)}

12. Shapiro, JH: Composition Operators and Classical Function Theory. Universitext: Tracts in Mathematics. Springer, New York (1993)

13. Singh, RK, Manhas, JS: Composition Operators on Function Spaces. North-Holland Mathematics Studies, vol. 179. North-Holland, Amsterdam (1993)

14. Nordgren, EA: Composition operators. Can. J. Math.20, 442-449 (1968) 15. Parrot, SK: Weighted translation operators. Thesis, University of Michigan (1965)

16. Singh, RK: Compact and quasinormal composition operators. Proc. Am. Math. Soc.45, 80-82 (1974) 17. Singh, RK: Composition operators induced by rational functions. Proc. Am. Math. Soc.59, 329-333 (1976) 18. Singh, RK, Chandra Kumar, RD: Compact weighted composition operators onL2₍_λ_{). Acta Sci. Math.}_{49, 339-344}

(1985)

19. Singh, RK, Komal, BS: Composition operator onp_{and its adjoint. Proc. Am. Math. Soc.}_{70, 21-25 (1978)}

20. Singh, RK, Kumar, A: Compact composition operators. J. Aust. Math. Soc. Ser. A28, 309-314 (1979) 21. Takagi, H: Compact weighted composition operators onLp_{. Proc. Am. Math. Soc.}_{116, 505-511 (1992)}

22. Takagi, H, Yokouchi, K: Multiplication and composition operators between twoLp_{-spaces. In: Function Spaces,}

Edwardsville, IL, 1998. Contemp. Math., vol. 232, pp. 321-338. Am. Math. Soc., Providence (1999) 23. Xu, XM: Compact composition operators onLP₍_X_,_A_,_μ_{). Adv. Math.}_{20(2), 221-225 (1991) (in Chinese)}

24. Cui, Y, Hudzik, H, Kumar, R, Maligranda, L: Composition operators in Orlicz spaces. J. Aust. Math. Soc.76(2), 189-206 (2004)

25. Gupta, S, Komal, BS, Suri, N: Weighted composition operators on Orlicz spaces. Int. J. Contemp. Math. Sci.5(1-4), 11-20 (2010)

26. Komal, BS, Gupta, S: Composition operators on Orlicz spaces. Indian J. Pure Appl. Math.32(7), 1117-1122 (2001) 27. Kumar, R: Composition operators on Orlicz spaces. Integral Equ. Oper. Theory29(1), 17-22 (1997)

28. Raj, K, Kumar, A, Sharma, SK: Composition operators on Musielak-Orlicz spaces. J. Contemp. Appl. Math.1(1), 54-62 (2011)

29. Raj, K, Sharma, AK, Kumar, A: Weighted composition operators on Musielak-Orlicz spaces. Ars Comb.107, 431-439 (2012)

30. Raj, K, Jamwal, S, Sharma, SK: Weighted composition operators on Musielak-Orlicz spaces of Bochner-type. Ann. Univ. Buchar. Math. Ser.4(LXII)(2), 513-523 (2013)

31. Raj, K, Sharma, SK: Composition operators on Musielak-Orlicz spaces of Bochner type. Math. Bohem.137(4), 449-457 (2012)

32. Arora, SC, Datt, G: Compact weighted composition operators on Orlicz-Lorentz spaces. Acta Sci. Math.77(3-4), 567-578 (2011)

33. Arora, SC, Datt, G, Verma, S: Multiplication and composition operators on Orlicz-Lorentz spaces. Int. J. Math. Anal. 1(25-28), 1227-1234 (2007)

34. Castillo, RE, Chaparro, HC, Ramos Fernández, JC: Orlicz-Lorentz spaces and their composition operators. Proyecciones 34(1), 85-105 (2015)

(18)

36. Cui, Y, Hudzik, H, Kumar, R, Kumar, R: Erratum and addendum to the paper ‘Compactness and essential norms of composition operators on Orlicz-Lorentz spaces’ [Applied Mathematics Letters 25 (2012) 1778-1783]. Comment. Math.54(2), 259-260 (2014)

37. Kumar, R, Kumar, R: Composition operators on Orlicz-Lorentz spaces. Integral Equ. Oper. Theory60(1), 79-88 (2008) 38. Nordgren, EA: Composition Operators on Hilbert Spaces. Lecture Notes in Mathematics, vol. 693, pp. 37-68. Springer,

New York (1978)

39. Singh, RK, Chandra Kumar, RD: Weighted composition operators on functional Hilbert spaces. Bull. Aust. Math. Soc. 31(1), 117-126 (1985)

40. Al-Rawashdeh, W: Compact composition operators on weighted Hilbert spaces. J. Appl. Funct. Anal.10(1-2), 101-108 (2015)

41. Boyd, DM: Composition operators on the Bergman spaces. Colloq. Math.34, 127-136 (1975) 42. Boyd, DM: Composition operators onHP₍_A_{). Pac. J. Math.}_{62, 55-60 (1976)}

43. Charpentier, S: Composition operators on weighted Bergman-Orlicz spaces on the ball. Complex Anal. Oper. Theory 7(1), 43-68 (2013)

44. Komal, BS, Gupta, S: Multiplication operators between Orlicz spaces. Integral Equ. Oper. Theory41(3), 324-330 (2001) 45. Raj, K, Sharma, SK, Kumar, A: Multiplication operators on Musielak-Orlicz spaces of Bochner type. J. Adv. Stud. Topol.

3(4), 1-7 (2012)

46. Seid, HA: Cyclic multiplication operators onLp_{-spaces. Pac. J. Math.}_{51, 542-562 (1974)}

47. Sharma, SK, Komal, BS: Multiplication operators on weighted Hardy spaces. Lobachevskii J. Math.33(2), 165-169 (2012)

48. Singh, RK, Kumar, A: Multiplication operators and composition operators with closed ranges. Bull. Aust. Math. Soc.16, 247-252 (1977)

49. Singh, RK, Manhas, JS: Multiplication operators on weighted spaces of vector-valued continuous functions. J. Aust. Math. Soc. Ser. A50, 98-107 (1991)

50. Hudzik, H, Krbec, M: On non-eﬀective weights in Orlicz spaces. Indag. Math.18(2), 215-231 (2007)

51. Musielak, J: Orlicz Spaces and Modular Spaces. Lecture Notes in Mathematics, vol. 1034. Springer, Berlin (1983) 52. Musielak, J, Orlicz, W: On modular spaces. Stud. Math.18, 49-65 (1959)

53. Krasnoselskii, MA, Rutickii, YB: Convex Functions and Orlicz Spaces. Translation. Nordhoﬀ, Groningen (1961) 54. Luxemburg, WAJ: Banach function spaces. PhD thesis, Delft (1955)

55. Maligranda, L: Orlicz Spaces and Interpolation. Seminars in Math. 5. Univ. Estadual de Campinas, Campinas SP, Brazil (1989)

56. Ishii, J: On equivalence of modular function spaces. Proc. Jpn. Acad.35, 551-556 (1959)

57. Hudzik, H, Musielak, J, Urba ´nski, R: Interpolation of compact sublinear operators in generalized Orlicz space of nonsymmetric type. In: Topology. Vol. I, Proc. Fourth Colloq., Budapest, 1978. Colloq. Math. Soc. János Bolyai, vol. 23, pp. 625-638. North-Holland, Amsterdam (1980)