#### Abstract

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 be a -finite measure space and let denote the set of all equivalence classes of complex valued measurable functions defined on , where or . Then, for , the Cesàro function space is denoted by and is defined as The Cesàro function space is a Banach space under the norm see [1].

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

Let be a nonsingular measurable transformation; that is, , for each , whenever . This condition means that the measure is absolutely continuous with respect to . Let be the Radon-Nikodym derivative. In addition, we assume that is almost everywhere finite valued or equivalently that is -finite. An atom of the measure is an element with such that, for each , if , then either or . Let be an atom. Since is -finite, it follows that . Also every -measurable function on is constant almost everywhere on . It is a well-known fact that every sigma finite measure space can be decomposed into two disjoint sets and such that is atomic over and is a countable collection of disjoint atoms (see [9]).

Any nonsingular measurable transformation induces a linear operator from into the linear space of equivalence classes of -measurable functions on defined by , . Hence, the nonsingularity of guarantees that the operator is well defined. If takes into itself, then we call that is a composition operator on . By , we denote the set of all bounded linear operators from 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 and a number such that for every , in an appropriate domain, whenever the function is given. If varies in the open unit disk and 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 -finite subalgebra , there exists an operator , which is called conditional expectation operator; on the set of all nonnegative measurable functions or for each , the operator is uniquely determined by the following conditions:(i) is -measurable;(ii)if is any -measurable set for which exists, we have .The expectation operator has the following properties:(a); (b)if almost everywhere, then almost everywhere;(c);(d) has the form for exactly one -measurable function , in particular, is a well-defined measurable function;(e); this is a Cauchy-Schwartz inequality for conditional expectation;(f)for almost everywhere, almost everywhere.For deeper study of properties of , see [28].

Let be a Banach space and let be the set of all compact operators on . For , the Banach algebra of all bounded linear operators on into itself, the essential norm of means the distance from to in the operator norm; namely, Clearly, is compact if and only if . 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 under certain conditions on and .

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 be a -finite measure space and let be nonsingular measurable transformation. Then, induces a composition operator on if and only if there exists such that for every . Moreover,
*

*Proof. *Suppose that is a composition operator. If such that , then and

Let . Then,
Conversely, suppose that the condition is true. Then, and hence the Radon-Nikodym derivative of with respect to exists and a.e.

Let . Then,
Therefore, .

Now, Let . Then, for all and . Thus for all . By the first part of the theorem, we have

Hence, . Thus,
On the other hand, Let . Thus, for all . In particular, for , such that , we have and .

Therefore,
From (10) and (11), we obtain

Theorem 2. *If is a linear transformation, then is continuous.*

*Proof. *Let and be sequences in such that
Then, we can find a subsequence of such that

From the nonsingularity of ,
Then, from (13) and (15), we conclude that . This proves that graph of is closed and hence, by closed graph theorem, 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 . Then, is compact if and only if is finite dimensional, for each , where
*

*Proof. *For , we have
Then, is compact if and only if is a compact operator if and only if is a finite dimensional, where is the identity operator.

Corollary 4. *If is a nonatomic measure space. Then no nonzero composition operator on is compact.*

Let be the decomposition of into nonatomic and atomic parts, respectively. If or and consists of finitely many atoms, then, by Theorem 3, does not admit a nonzero compact composition operator. Thus, in this case, and hence .

Now, we present the main result of this section.

Theorem 5. *Let consists of finitely many atoms. Suppose that is a finite dimensional; that is, is a finite dimensional. Let . Then,*(i)* iff ;*(ii)* if ;*(iii)* if .*

*Proof. *(i) Theorem 3 implies that is compact if and only if . So (i) is the direct consequence of Theorem 3.

(ii) Suppose that . Take arbitrary. The definition of implies that either contains a nonatomic subset or has infinitely many atoms. If contains a nonatomic subset, then there are measurable sets such that . Define . Then, , for all . We claim that weakly. For this, we show that , for all . Let with and . Then, we have
Since simple functions are dense in , thus is proved to converge to zero weakly. Now, assume that consists of infinitely many atoms. Let be disjoint atoms in . Again, put as above. It is easy to see that, for with , we have for sufficiently large . So, in both cases, . Now, we claim that . Since , we see that
Finally, take a compact operator on such that . Then, we have
for all . Since a compact operator maps weakly convergent sequences into norm convergent ones, it follows . Hence, . Since is arbitrary, we obtain .

(iii) Let and take arbitrary. Put . The definition of implies that consists of finitely many atoms. So, we can write , where are distinct. Since , for all , hence has finite rank. Now, let such that . Then, we have
It follows that . Since simple functions are dense in . We obtain
Finally, Since is a compact operator, we get
It follows that and, consequently, .

*Example 6. *Let , where is the set of natural numbers. Let be the Lebesgue measure on and if . Define as , and .

Consider , for , and , for all . Then, we can easily calculate on for .

#### 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 and such that , then We find that every gives rise to a bounded linear functional which is defined as For each , there exists a unique measurable function such that for measurable function for which the left integral exists. The function is called conditional expectation of with respect to the -algebra . The operator defined by is called the Frobenius Perron operator where if and only if .

Theorem 7. *Let . Then has closed range if and only if there exists such that for -almost all .*

*Proof. *If for -almost all , then, for min,
Hence, , for all , so that has closed range.

Conversely, suppose that has closed range. Then there exists such that
for every . Choose a positive integer such that . If the set has positive measure, then, for a given measurable subset supp such that , we have
or equivalently
This contradicts inequality (27). Hence, is bounded away from zero on supp.

Theorem 8. *Let . Then, is either zero-dimensional or infinite dimensional.*

*Proof. *Suppose ker . Then supp is a set of nonzero measure. Now we can partition into a sequence of measurable sets, . We show that ker . Consider
Hence, if ker is not zero-dimensional, it is infinite dimensional.

Corollary 9. *Let . Then, is injective if and only if is surjective.*

Theorem 10. *Let . Then, has dense range if and only if .*

*Proof. *Suppose that has dense range. Let such that . Then, there exists such that . Now, we can find a subsequence of such that a.e. Now, each is measurable with respect to . Therefore, is measurable with respect to so that . Hence, a.e.

Conversely, suppose a.e. If , then there exists such that . Since is -finite, we can find an increasing sequence of sets of finite measure or . Hence, for given , there exists a positive integer such that for every . Hence,
for all . Then . This proves that has dense range.

Theorem 11. *Let . Then, is Fredholm if and only if is invertible.*

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

Corollary 12. *Let . Then, is an isometry if and only if is measure preserving.*

*Proof. *If is measure preserving, then a.e. Therefore,
Hence, is an isometry. Conversely, if is an isometry, then
This implies that . Hence, a.e.

Theorem 13. *Let . Then, .*

*Proof. *Let such that . For ,
Hence, . After identifying with , we can write .

#### Conflict of Interests

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