The compact, invertible, Fredholm, and closed range composition operators are characterized. We also make an effort to compute the essential norm of composition operators on the Cesàro function spaces.

1. Introduction and Preliminaries

Let (X,s,μ) be a σ-finite measure space and let L0=L0(X) denote the set of all equivalence classes of complex valued measurable functions defined on X, where X=[0,1] or X=[0,∞). Then, for 1≤p<∞, the Cesàro function space is denoted by Cesp(X) and is defined as
(1)Cesp(X)={f∈L0(X):∫X(1x∫0x|f(t)|dμ(t))pdμ(x)<∞}.
The Cesàro function space Cesp(X) is a Banach space under the norm
(2)∥f∥=(∫X(1x∫0x|f(t)|dμ(t))pdμ(x))1/p;
see [1].

The Cesàro functions spaces Cesp[0,∞) for 1≤p≤∞ were considered by Shiue [2], Hassard and Hussein [3], and Sy et al. [4]. The space Ces∞[0,1] appeared already in 1948 and it is known as the Korenblyum, Krein, and Levin space K (see [5, 6]). Recently, in [7], it is proved that, in contrast to Cesàro sequence spaces, the Cesàro function spaces Cesp(X) on both X=[0,1] and X=[0,∞) for 1<p<∞ are not reflexive and they do not have the fixed point property. In [8], Astashkin and Maligranda investigated Rademacher sums in Cesp[0,1] for 1≤p≤∞. The description is different for 1≤p<∞ and p=∞.

Let T:X→X be a nonsingular measurable transformation; that is, μT-1(A)=μ(T-1(A))=0, for each A∈s, whenever μ(A)=0. This condition means that the measure μT-1 is absolutely continuous with respect to μ. Let f0=dμT-1/dμ be the Radon-Nikodym derivative. In addition, we assume that f0 is almost everywhere finite valued or equivalently that (X,T-1(s),μ) is σ-finite. An atom of the measure μ is an element A∈s with μ(A)>0 such that, for each F∈s, if F⊂A, then either μ(F)=0 or μ(F)=μ(A). Let A be an atom. Since μ is σ-finite, it follows that μ(A)<∞. Also every s-measurable function f on X is constant almost everywhere on A. It is a well-known fact that every sigma finite measure space (X,s,μ) can be decomposed into two disjoint sets X1 and X2 such that μ is atomic over X1 and X2 is a countable collection of disjoint atoms (see [9]).

Any nonsingular measurable transformation T induces a linear operator CT from Cesp(X) into the linear space of equivalence classes of s-measurable functions on X defined by CTf=f∘T, f∈Cesp(X). Hence, the nonsingularity of T guarantees that the operator CT is well defined. If CT takes Cesp(X) into itself, then we call that CT is a composition operator on Cesp(X). By B(Cesp(X)), we denote the set of all bounded linear operators from Cesp(X) into itself.

So far as we know, the earliest appearance of a composition transformation was in 1871 in a paper of Schrljeder [10], where it is asked to find a function f and a number α such that
(3)(f∘T)(z)=αf(z),
for every z, in an appropriate domain, whenever the function T is given. If z varies in the open unit disk and T is an analytic function, then a solution is obtained by Köenigs [11]. In 1925, these operators were employed in Littlewood’s subordination theory [12]. In the early 1930s, the composition operators were used to study problems in mathematical physics and especially classical mechanics; see Koopman [13]. In those days, these operators were known as substitution operators. The systematic study of composition operators has relatively a very short history. It was started by Nordgren in 1968 in his paper [14]. After this, the study of composition operators has been extended in several directions by several mathematicians. For more details on composition operators, see [15–27] and references therein.

Associated with each σ-finite subalgebra s0⊂s, there exists an operator E=Es0, which is called conditional expectation operator; on the set of all nonnegative measurable functions f or for each f∈L0(X,s,μ), the operator E is uniquely determined by the following conditions:

E(f) is s0-measurable;

if A is any s0-measurable set for which ∫Afdμ exists, we have ∫Afdμ=∫AE(f)dμ.

The expectation operator E has the following properties:

E(f·g∘T)=E(f)·(g∘T);

if f≥g almost everywhere, then E(f)≥E(g) almost everywhere;

E(1)=1;

E(f) has the form E(f)=g∘T for exactly one σ-measurable function g, in particular, g=E(f)∘T-1 is a well-defined measurable function;

|E(fg)|2≤(E|f|2)(E|g|2); this is a Cauchy-Schwartz inequality for conditional expectation;

for f>0 almost everywhere, E(f)>0 almost everywhere.

For deeper study of properties of E, see [28].

Let ℬ be a Banach space and let 𝒦 be the set of all compact operators on ℬ. For U∈ℒ(ℬ), the Banach algebra of all bounded linear operators on ℬ into itself, the essential norm of U means the distance from U to 𝒦 in the operator norm; namely,
(4)∥U∥e=inf{∥U-S∥:S∈𝒦}.
Clearly, U is compact if and only if ∥U∥e=0. As seen in [29], the essential norm plays an interesting role in the compact problems of concrete operators. Many people have computed the essential norm of various concrete operators. For the study of essential norm of composition operators, see [30–33] and reference therein.

The question of actually calculating the norm and essential norm of composition operators on Cesàro function spaces is not a trivial one. In spite of the difficulties associated with computing the essential norm exactly, it is often possible to find upper and lower bound for the essential norm of CT:Cesp(X)→Cesp(X) under certain conditions on p and X.

The main purpose of this paper is to characterize the boundedness, compactness, closed range, and Fredholmness of composition operators on Cesàro function spaces. We also make an effort to compute the essential norm of composition operators in Section 3 of this paper.

2. Composition Operators

In this section of the paper, we will investigate the necessary and sufficient condition for a composition operator to be bounded.

Theorem 1.

Let (X,s,μ) be a σ-finite measure space and let T:X→X be nonsingular measurable transformation. Then, T induces a composition operator CT on Cesp(X) if and only if there exists M>0 such that μT-1(E)≤Mμ(E) for every E∈s. Moreover,
(5)∥CT∥=sup0<μ(E)<∞((μ(T-1(E))μ(E))p)1/p.

Proof.

Suppose that CT is a composition operator. If E∈s such that μ(E)<∞, then χE∈Cesp(X) and
(6)μT-1(E)=∥CTχE∥p≤∥CT∥p∥χE∥p=∥CT∥pμ(E).

Let M=∥CT∥p. Then,
(7)μT-1(E)≤Mμ(E).
Conversely, suppose that the condition is true. Then, μT-1≪μ and hence the Radon-Nikodym derivative f0 of μT-1 with respect to μ exists and f0≤M a.e.

Let f∈Cesp(X). Then,
(8)∥CTf∥p=∫X(1x∫0x|(f∘T)(t)|dμ(t))pdμ(x)≤∫X(1x∫0x|f|dμT-1(t))pdμ(x)=∫X(1x∫0x|f|f0dμ(t))pdμ(x)≤Mp∫X(1x∫0x|f(t)|dμ(t))pdμ(x).
Therefore, ∥CTf∥≤M∥f∥.

Now, Let N=sup0<μ(E)<∞((μ(T-1(E))/μ(E))p)1/p. Then, ((μ(T-1(E))/μ(E))p)1/p≤N for all E∈s and μ(E)≠0. Thus (μT-1(E))p≤Np(μ(E))p for all E∈s. By the first part of the theorem, we have
(9)∥CTf∥≤N∥f∥,∀f∈Cesp(X).

Hence, ∥CT∥=supf≠0∥CTf∥/∥f∥≤N. Thus,
(10)∥CT∥≤sup0<μ(E)<∞((μ(T-1(E))μ(E))p)1/p.
On the other hand, Let M=∥CT∥=supf≠0∥CTf∥/∥f∥. Thus, ∥CTf∥/∥f∥≤M for all f∈Cesp(X),f≠0. In particular, for f=χE, such that 0<μ(E)<∞,E∈s, we have f=χE∈Cesp(X) and ∥CTχE∥/∥χE∥=((μ(T-1(E))/μ(E))p)1/p≤M.

Therefore,
(11)sup0<μ(E)<∞((μ(T-1(E))μ(E))p)1/p≤M=∥CT∥.
From (10) and (11), we obtain
(12)∥CT∥=sup0<μ(E)<∞((μ(T-1(E))μ(E))p)1/p.

Theorem 2.

If CT:Cesp(X)→Cesp(X) is a linear transformation, then CT is continuous.

Proof.

Let {fn} and {CTfn} be sequences in Cesp(X) such that
(13)fn⟶f,CTfn⟶gforsomef,g∈Cesp(X).
Then, we can find a subsequence {fnk} of {fn} such that
(14)∥fnk-f∥(t)⟶0forμ-almostallt∈X.

From the nonsingularity of T,
(15)(∥fnk-f∥∘T)(t)⟶0forμ-almostallt∈X.
Then, from (13) and (15), we conclude that CTf=g. This proves that graph of CT is closed and hence, by closed graph theorem, CT is continuous.

3. Compactness and Essential Norm of Composition Operators

This section is devoted to the study of compact composition operators on Cesàro function spaces. A necessary and sufficient condition for a composition operator to be compact is reported in this section. The main aim of this section is to compute the essential norm of the composition operators.

Theorem 3.

Let CT∈B(Cesp(X)). Then, CT is compact if and only if Cesp(Xϵ,μT-1) is finite dimensional, for each ϵ>0, where
(16)Xϵ={x∈X:dμT-1dμ(x)≥ϵ}.

Proof.

For f∈Cesp(X), we have
(17)∥CTf∥μ=(∫X(1x∫0x|f∘T|(t)dμ(t))pdμ(x))1/p=(∫X(1x∫0x|f|dμT-1(t))pdμ(x))1/p=(∫X(1x∫0x|f|dμT-1(t))pdμT-1(x))1/p=∥f∥μT-1=∥If∥μT-1.
Then, CT is compact if and only if I:Cesp(Xϵ,μT-1)→Cesp(Xϵ,μT-1) is a compact operator if and only if Cesp(Xϵ,μT-1) is a finite dimensional, where I is the identity operator.

Corollary 4.

If (X,s,μ) is a nonatomic measure space. Then no nonzero composition operator on Cesp(X) is compact.

Let X=X1∪X2 be the decomposition of X into nonatomic and atomic parts, respectively. If X2=ϕ or μ(X)=+∞ and X2 consists of finitely many atoms, then, by Theorem 3, Cesp(X) does not admit a nonzero compact composition operator. Thus, in this case, 𝒦={0} and hence ∥CT∥e=∥CT∥.

Now, we present the main result of this section.

Theorem 5.

Let X2 consists of finitely many atoms. Suppose that Cesp(χϵ,μT-1) is a finite dimensional; that is, Xϵ={x∈X:(dμT-1/dμ)(x)≥ϵ} is a finite dimensional. Let α=inf{ϵ>0:Xϵisafinitedimensional}. Then,

∥CT∥e=0 iff α=0;

∥CT∥e≥α if 0<α≤1;

∥CT∥e≤α if α>1.

Proof.

(i) Theorem 3 implies that CT is compact if and only if α=0. So (i) is the direct consequence of Theorem 3.

(ii) Suppose that 0<α≤1. Take 0<ϵ<2α arbitrary. The definition of α implies that F=χα-(ϵ/2) either contains a nonatomic subset or has infinitely many atoms. If F contains a nonatomic subset, then there are measurable sets En,n∈ℕ such that En+1⊆En⊆F,0<μ(En)<1/n. Define fn=1/(μ(En))1/pχEn. Then, ∥fn∥=1, for all n∈ℕ. We claim that fn→0 weakly. For this, we show that ∫Xfngdμ→0, for all g∈(Cesq(X)). Let A⊆F with μ(A)<∞ and g=χA. Then, we have
(18)|∫XfnχAdμ|=1(μ(En))1/pμ(A∩En)≤(1n)1-(1/p)⟶0asn⟶∞.
Since simple functions are dense in Cesq(X), thus fn is proved to converge to zero weakly. Now, assume that F consists of infinitely many atoms. Let {En}n=0∞ be disjoint atoms in F. Again, put fn as above. It is easy to see that, for A⊆F with 0<μ(A)<∞, we have μ(A∩En)=0 for sufficiently large n. So, in both cases, ∫Xfngdμ→0. Now, we claim that ∥CTfn∥≥α-(ϵ/2). Since 0<α-(ϵ/2)<1, we see that
(19)∥CTfn∥=(∫X(1x∫0x|CTfn|dμ(t))pdμ(x))1/p=(∫X(1x∫0x|fn|f0(t)dμ(t))pdμ(x))1/p≥(∫X(1x∫0x(α-ϵ2)|fn|dμ(t))pdμ(x))1/p=(α-ϵ2)(∫X(1x∫0x|fn|dμ(t))pdμ(x))1/p=α-ϵ2.
Finally, take a compact operator S on Cesp(X) such that ∥CT-S∥≤∥CT∥e+(ϵ/2). Then, we have
(20)∥CT∥e>∥CT-S∥-ϵ2≥∥CTfn-Sfn∥-ϵ2≥∥CTfn∥-∥Sfn∥-ϵ2≥(α-ϵ2)-∥Sfn∥-ϵ2,
for all n∈ℕ. Since a compact operator maps weakly convergent sequences into norm convergent ones, it follows ∥Sfn∥→0. Hence, ∥CT∥e≥α-ϵ. Since ϵ is arbitrary, we obtain ∥CT∥e≥α.

(iii) Let α>1 and take ϵ>0 arbitrary. Put K=Xα+ϵ. The definition of α implies that K consists of finitely many atoms. So, we can write K=(E1,E2,…,Em), where E1,E2,…,Em are distinct. Since (LχKCTf)X=∑i=1mχK(Ei)f(T(Ei)), for all f∈Cesp(X), hence LχKCT has finite rank. Now, let F⊆X∖K such that 0<μ(F)<∞. Then, we have
(21)μT-1(F)≤(α+ϵ)μ(F).
It follows that ∥χF∘T∥≤(α+ϵ)∥χF∥. Since simple functions are dense in Cesp(X). We obtain
(22)sup∥f∥≤1∥χX∖Kf∘T∥≤sup∥f∥≤1∥χX∖Kf∥≤α+ϵ.
Finally, Since LχKCT is a compact operator, we get
(23)∥CT-LχKCT∥=sup∥f∥≤1∥(1-χK)CTf∥=sup∥f∥≤1∥χX∖KCTf∥≤α+ϵ.
It follows that ∥CT∥e≤α+ϵ and, consequently, ∥CT∥e≤α.

Example 6.

Let X=(-∞,0]∪ℕ, where ℕ is the set of natural numbers. Let μ be the Lebesgue measure on (-∞,0] and μ({n})=1/2n if n∈ℕ. Define T:ℕ→ℕ as T(1)=T(2)=T(3)=1,T(4)=2,T(5)=T(6)=3, and T(2n+1)=5,forn≥3.

Consider T(2n)=2n-2, for n≥4, and T(x)=5x, for all x∈(-∞,0]. Then, we can easily calculate ∥CT∥e=3-1/p on Cesp(X) for 1<p<∞.

4. Fredholm and Isometric Composition Operators

In this section, we first establish a condition for the composition operators to have closed range and then we make the use of it to characterize the Fredholm composition operators. We also make an attempt to compute the adjoint of the composition operators.

Holder’s inequality for Cesàro measurable function spaces is that, if f∈Cesp(X) and g∈Cesq(X) such that 1/p+1/q=1, then
(24)∫fgdμ≤∥f∥p∥g∥q.
We find that every g∈Cesq(X) gives rise to a bounded linear functional Fg∈(Cesp(X))* which is defined as
(25)Fg(f)=∫fgdμ,foreveryf∈Cesp(X).
For each f∈Cesp(X), there exists a unique T-1(s) measurable function E(f) such that ∫gfdμ=∫gE(f)dμ for T-1(s) measurable function g for which the left integral exists. The function E(f) is called conditional expectation of f with respect to the σ-algebra T-1(s). The operator PT:Cesp(X)→Cesp(X) defined by PTf=f0E(f)∘T-1 is called the Frobenius Perron operator where E(f)∘T-1=g if and only if E(f)=g∘T.

Theorem 7.

Let CT∈B(Cesp(X)). Then CT has closed range if and only if there exists δ>0 such that f0(x)≥δ for μ-almost all x∈suppf0=S.

Proof.

If f0(x)≥δ for μ-almost all x∈S, then, for η= min(δ,1/δ)≤1,
(26)1≥(∫X(1x∫0x|CTf|∥CTf∥dμ(t))pdμ(x))1/p=(∫X(1x∫0xf0|f|∥CTf∥dμ(t))pdμ(x))1/p≥(∫X(1x∫0xη|f|∥CTf∥dμ(t))pdμ(x))1/p.
Hence, ∥CTf∥≥η∥f∥, for all f∈Cesp(S), so that CT has closed range.

Conversely, suppose that CT has closed range. Then there exists δ≥0 such that
(27)∥CTf∥≥δ∥f∥,
for every f∈Cesp(S). Choose a positive integer n such that 1/n<δ. If the set E={x∈X:f0(x)<1/n} has positive measure, then, for a given measurable subset F⊂suppf0 such that 0<μ(F)<∞, we have
(28)μT-1(E)<1nμ(E)
or equivalently
(29)∥CTχE∥≤1n∥χE∥.
This contradicts inequality (27). Hence, f0 is bounded away from zero on suppf0.

Theorem 8.

Let CT∈B(Cesp(X)). Then, kerCT* is either zero-dimensional or infinite dimensional.

Proof.

Suppose 0≠g∈ ker CT*. Then E= supp g is a set of nonzero measure. Now we can partition E into a sequence {En} of measurable sets, 0<μ(En)<∞. We show that gχEn∘T∈ ker CT*. Consider
(30)CT*(gχEn∘T)(f)=∫E(gχEn∘T)(CTf)dμ=∫EgCT(χEnf)dμ=0.
Hence, if ker CT* is not zero-dimensional, it is infinite dimensional.

Corollary 9.

Let CT∈B(Cesp(X)). Then, CT is injective if and only if T is surjective.

Theorem 10.

Let CT∈B(Cesp(X)). Then, CT has dense range if and only if T-1(s)=s.

Proof.

Suppose that CT has dense range. Let E∈s such that χE∈Cesp(X). Then, there exists {fn}⊂Cesp(X) such that CTfn→χE. Now, we can find a subsequence {fnk} of {fn} such that CTfnk→χE a.e. Now, each CTfnk is measurable with respect to T-1(s). Therefore, χE is measurable with respect to T-1(s) so that χE=χT-1(F). Hence, T-1(s)=s a.e.

Conversely, suppose T-1(s)=s a.e. If E∈s,0<μ(E)<∞, then there exists F∈s such that μ(T-1(F)ΔE)=0. Since X is σ-finite, we can find an increasing sequence {Fn} of sets of finite measure Fn↑F or T-1(F)∖T-1(Fn)↓0. Hence, for given ϵ>0, there exists a positive integer n0 such that μ(T-1(F∖Fn))<ϵ for every n≥n0. Hence,
(31)∥CTχF-CTχFn∥=∥CT(χF∖Fn)∥=∥χT-1(F∖Fn)∥=(∫X(1x∫0xχT-1(F∖Fn)dμ(t))pdμ(x))p<ϵ,
for all n≥n0. Then χF=χT-1(F)∈ranCT¯. This proves that CT has dense range.

Theorem 11.

Let CT∈B(Cesp(X)). Then, CT is Fredholm if and only if CT is invertible.

Proof.

Assume that CT is Fredholm. In view of Theorem 8, ker CT and ker CT* are zero-dimensional so that CT is injective and T-1(s)=s a.e. Therefore, by Theorem 10, CT has dense range. Since ranCT is closed, so CT is surjective. This proves the invertibility of CT. The proof of the converse part is obvious.

Corollary 12.

Let CT∈B(Cesp(X)). Then, CT is an isometry if and only if T is measure preserving.

Proof.

If T is measure preserving, then f0=1 a.e. Therefore,
(32)∥CTf∥=(∫X(1x∫0x|f∘T|(t)dμ(t))pdμ(x))1/p=(∫X(1x∫0xf0|f|dμ(t))pdμ(x))1/p=(∫X(1x∫0x|f|dμ(t))pdμ(x))1/p=∥f∥.
Hence, CT is an isometry. Conversely, if CT is an isometry, then
(33)∥CTχE∥=∥χE∥.
This implies that μ(T-1(E))=μ(E)forE∈s. Hence, f0=1 a.e.

Theorem 13.

Let CT∈B(Cesp(X)). Then, CT*=PT.

Proof.

Let A∈s such that 0<μ(A)<∞. For g∈Cesq(X),
(34)(CT*Fg)(χA)=Fg(CTχA)=∫CTχAgdμ=∫χA∘Tgdμ=∫χAE(g)∘T-1f0dμ=FE(g)∘T-1f0(χA).
Hence, CT*Fg=FE(g)∘T-1f0. After identifying g∈Cesq(X) with Fg∈(Cesp(X))*, we can write CT*g=E(g)∘T-1f0=PTg.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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