JFS Journal of Function Spaces 2314-8888 2314-8896 Hindawi Publishing Corporation 501057 10.1155/2014/501057 501057 Research Article Composition Operators on Cesàro Function Spaces http://orcid.org/0000-0002-2611-3391 Raj Kuldip Pandoh Suruchi Jamwal Seema Saejung Satit 1 School of Mathematics Shri Mata Vaishno Devi University Katra 182320 India smvdu.net.in 2014 3012014 2014 17 05 2013 20 11 2013 30 1 2014 2014 Copyright © 2014 Kuldip Raj et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 L 0 = L 0 ( 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 Ce s p ( X ) and is defined as (1) Ce s p ( X ) = { f L 0 ( X ) : X ( 1 x 0 x | f ( t ) | d μ ( t ) ) p d μ ( x ) < } . The Cesàro function space Ce s p ( X ) is a Banach space under the norm (2) f = ( X ( 1 x 0 x | f ( t ) | d μ ( t ) ) p d μ ( x ) ) 1 / p ; see .

The Cesàro functions spaces Ce s p [ 0 , ) for 1 p were considered by Shiue , Hassard and Hussein , and Sy et al. . The space Ce s [ 0,1 ] appeared already in 1948 and it is known as the Korenblyum, Krein, and Levin space K (see [5, 6]). Recently, in , it is proved that, in contrast to Cesàro sequence spaces, the Cesàro function spaces Ce s p ( 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 , Astashkin and Maligranda investigated Rademacher sums in Ce s p [ 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, μ T - 1 ( A ) = μ ( T - 1 ( A ) ) = 0 , for each A s , whenever μ ( A ) = 0 . This condition means that the measure μ T - 1 is absolutely continuous with respect to μ . Let f 0 = d μ T - 1 / d μ be the Radon-Nikodym derivative. In addition, we assume that f 0 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 X 1 and X 2 such that μ is atomic over X 1 and X 2 is a countable collection of disjoint atoms (see ).

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

So far as we know, the earliest appearance of a composition transformation was in 1871 in a paper of Schrljeder , where it is asked to find a function f and a number α such that (3) ( f T ) ( z ) = α f ( 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 . In 1925, these operators were employed in Littlewood’s subordination theory . In the early 1930s, the composition operators were used to study problems in mathematical physics and especially classical mechanics; see Koopman . 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 . After this, the study of composition operators has been extended in several directions by several mathematicians. For more details on composition operators, see  and references therein.

Associated with each σ -finite subalgebra s 0 s , there exists an operator E = E s 0 , which is called conditional expectation operator; on the set of all nonnegative measurable functions f or for each f L 0 ( X , s , μ ) , the operator E is uniquely determined by the following conditions:

E ( f ) is s 0 -measurable;

if A is any s 0 -measurable set for which A f d μ exists, we have A f d μ = A E ( f ) d μ .

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 ( f g ) | 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 .

Let be a Banach space and let 𝒦 be the set of all compact operators on . For U ( ) , the Banach algebra of all bounded linear operators on 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 , 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  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 C T : Ce s p ( X ) Ce s p ( 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 C T on Ce s p ( X ) if and only if there exists M > 0 such that μ T - 1 ( E ) M μ ( E ) for every E s . Moreover, (5) C T = sup 0 < μ ( E ) < ( ( μ ( T - 1 ( E ) ) μ ( E ) ) p ) 1 / p .

Proof.

Suppose that C T is a composition operator. If E s such that μ ( E ) < , then χ E Ce s p ( X ) and (6) μ T - 1 ( E ) = C T χ E p C T p χ E p = C T p μ ( E ) .

Let M = C T p . Then, (7) μ T - 1 ( E ) M μ ( E ) . Conversely, suppose that the condition is true. Then, μ T - 1 μ and hence the Radon-Nikodym derivative f 0 of μ T - 1 with respect to μ exists and f 0 M a.e.

Let f Ce s p ( X ) . Then, (8) C T f p = X ( 1 x 0 x | ( f T ) ( t ) | d μ ( t ) ) p d μ ( x ) X ( 1 x 0 x | f | d μ T - 1 ( t ) ) p d μ ( x ) = X ( 1 x 0 x | f | f 0 d μ ( t ) ) p d μ ( x ) M p X ( 1 x 0 x | f ( t ) | d μ ( t ) ) p d μ ( x ) . Therefore, C T f M f .

Now, Let N = sup 0 < 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 ( μ T - 1 ( E ) ) p N p ( μ ( E ) ) p       for all    E s . By the first part of the theorem, we have (9) C T f N f , f Ce s p ( X ) .

Hence, C T = sup f 0 C T f / f N . Thus, (10) C T sup 0 < μ ( E ) < ( ( μ ( T - 1 ( E ) ) μ ( E ) ) p ) 1 / p . On the other hand, Let M = C T = sup f 0 C T f / f . Thus, C T f / f M for all f Ce s p ( X ) , f 0 . In particular, for f = χ E , such that 0 < μ ( E ) < , E s , we have f = χ E Ce s p ( X ) and C T χ E / χ E = ( ( μ ( T - 1 ( E ) ) / μ ( E ) ) p ) 1 / p M .

Therefore, (11) sup 0 < μ ( E ) < ( ( μ ( T - 1 ( E ) ) μ ( E ) ) p ) 1 / p M = C T . From (10) and (11), we obtain (12) C T = sup 0 < μ ( E ) < ( ( μ ( T - 1 ( E ) ) μ ( E ) ) p ) 1 / p .

Theorem 2.

If C T : Ce s p ( X ) Ce s p ( X ) is a linear transformation, then C T is continuous.

Proof.

Let { f n } and { C T f n } be sequences in Ce s p ( X ) such that (13) f n f , C T f n g for some f , g Ce s p ( X ) . Then, we can find a subsequence { f n k } of { f n } such that (14) f n k - f ( t ) 0 for μ -almost all t X .

From the nonsingularity of T , (15) ( f n k - f T ) ( t ) 0 for μ -almost all t X . Then, from (13) and (15), we conclude that C T f = g . This proves that graph of C T is closed and hence, by closed graph theorem, C T 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 C T B ( Ce s p ( X ) ) . Then, C T is compact if and only if Ce s p ( X ϵ , μ T - 1 ) is finite dimensional, for each ϵ > 0 , where (16) X ϵ = { x X : d μ T - 1 d μ ( x ) ϵ } .

Proof.

For f Ce s p ( X ) , we have (17) C T f μ = ( X ( 1 x 0 x | f T | ( t ) d μ ( t ) ) p d μ ( x ) ) 1 / p = ( X ( 1 x 0 x | f | d μ T - 1 ( t ) ) p d μ ( x ) ) 1 / p = ( X ( 1 x 0 x | f | d μ T - 1 ( t ) ) p d μ T - 1 ( x ) ) 1 / p = f μ T - 1 = I f μ T - 1 . Then, C T is compact if and only if I : Ce s p ( X ϵ , μ T - 1 ) Ce s p ( X ϵ , μ T - 1 ) is a compact operator if and only if Ce s p ( X ϵ , μ T - 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 Ce s p ( X ) is compact.

Let X = X 1 X 2 be the decomposition of X into nonatomic and atomic parts, respectively. If X 2 = ϕ or μ ( X ) = + and X 2 consists of finitely many atoms, then, by Theorem 3, Ce s p ( X ) does not admit a nonzero compact composition operator. Thus, in this case, 𝒦 = { 0 } and hence C T e = C T .

Now, we present the main result of this section.

Theorem 5.

Let X 2 consists of finitely many atoms. Suppose that Ce s p ( χ ϵ , μ T - 1 ) is a finite dimensional; that is, X ϵ = { x X : ( d μ T - 1 / d μ ) ( x ) ϵ } is a finite dimensional. Let α = inf { ϵ > 0 : X ϵ       i s a f i n i t e d i m e n s i o n a l } . Then,

C T e = 0   iff α = 0 ;

C T e α   if 0 < α 1 ;

C T e α   if α > 1 .

Proof.

(i) Theorem 3 implies that C T 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 < 2 α arbitrary. The definition of α implies that F = χ α - ( ϵ / 2 ) either contains a nonatomic subset or has infinitely many atoms. If F contains a nonatomic subset, then there are measurable sets E n , n such that E n + 1 E n F , 0 < μ ( E n ) < 1 / n . Define f n = 1 / ( μ ( E n ) ) 1 / p χ E n . Then, f n = 1 , for all n . We claim that f n 0 weakly. For this, we show that X f n g d μ 0 , for all g ( Ce s q ( X ) ) . Let A F with μ ( A ) < and g = χ A . Then, we have (18) | X f n χ A d μ | = 1 ( μ ( E n ) ) 1 / p μ ( A E n ) ( 1 n ) 1 - ( 1 / p ) 0 as n . Since simple functions are dense in Ce s q ( X ) , thus f n is proved to converge to zero weakly. Now, assume that F consists of infinitely many atoms. Let { E n } n = 0 be disjoint atoms in F . Again, put f n as above. It is easy to see that, for A F with 0 < μ ( A ) < , we have μ ( A E n ) = 0 for sufficiently large n . So, in both cases, X f n g d μ 0 . Now, we claim that C T f n α - ( ϵ / 2 ) . Since 0 < α - ( ϵ / 2 ) < 1 , we see that (19) C T f n = ( X ( 1 x 0 x | C T f n | d μ ( t ) ) p d μ ( x ) ) 1 / p = ( X ( 1 x 0 x | f n | f 0 ( t ) d μ ( t ) ) p d μ ( x ) ) 1 / p ( X ( 1 x 0 x ( α - ϵ 2 ) | f n | d μ ( t ) ) p d μ ( x ) ) 1 / p = ( α - ϵ 2 ) ( X ( 1 x 0 x | f n | d μ ( t ) ) p d μ ( x ) ) 1 / p = α - ϵ 2 . Finally, take a compact operator S on Ce s p ( X ) such that C T - S C T e + ( ϵ / 2 ) . Then, we have (20) C T e > C T - S - ϵ 2 C T f n - S f n - ϵ 2 C T f n - S f n - ϵ 2 ( α - ϵ 2 ) - S f n - ϵ 2 , for all n . Since a compact operator maps weakly convergent sequences into norm convergent ones, it follows S f n 0 . Hence, C T e α - ϵ . Since ϵ is arbitrary, we obtain C T e α .

(iii) Let α > 1 and take ϵ > 0 arbitrary. Put K = X α + ϵ . The definition of α implies that K consists of finitely many atoms. So, we can write K = ( E 1 , E 2 , , E m ) , where E 1 , E 2 , , E m are distinct. Since ( L χ K C T f ) X = i = 1 m χ K ( E i ) f ( T ( E i ) ) , for all f Ce s p ( X ) , hence L χ K C T has finite rank. Now, let F X K such that 0 < μ ( F ) < . Then, we have (21) μ T - 1 ( F ) ( α + ϵ ) μ ( F ) . It follows that χ F T ( α + ϵ ) χ F . Since simple functions are dense in Ce s p ( X ) . We obtain (22) sup f 1 χ X K f T sup f 1 χ X K f α + ϵ . Finally, Since L χ K C T is a compact operator, we get (23) C T - L χ K C T = sup f 1 ( 1 - χ K ) C T f = sup f 1 χ X K C T f α + ϵ . It follows that C T e α + ϵ and, consequently, C T e α .

Example 6.

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

Consider T ( 2 n ) = 2 n - 2 , for n 4 , and T ( x ) = 5 x , for all x ( - , 0 ] . Then, we can easily calculate C T e = 3 - 1 / p   on   Ce s p ( 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 Ce s p ( X ) and g Ce s q ( X ) such that 1 / p + 1 / q = 1 , then (24) f g d μ f p g q . We find that every g Ce s q ( X ) gives rise to a bounded linear functional F g ( Ce s p ( X ) ) * which is defined as (25) F g ( f ) = f g d μ , for every f Ce s p ( X ) . For each f Ce s p ( X ) , there exists a unique T - 1 ( s ) measurable function E ( f ) such that g f d μ = g E ( f ) d μ 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 P T : Ce s p ( X ) Ce s p ( X ) defined by P T f = f 0 E ( 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 C T B ( Ce s p ( X ) ) . Then C T has closed range if and only if there exists δ > 0 such that f 0 ( x ) δ for μ -almost all x supp f 0 = S .

Proof.

If f 0 ( x ) δ for μ -almost all x S , then, for η = min ( δ , 1 / δ ) 1 , (26) 1 ( X ( 1 x 0 x | C T f | C T f d μ ( t ) ) p d μ ( x ) ) 1 / p = ( X ( 1 x 0 x f 0 | f | C T f d μ ( t ) ) p d μ ( x ) ) 1 / p ( X ( 1 x 0 x η | f | C T f d μ ( t ) ) p d μ ( x ) ) 1 / p . Hence, C T f η f , for all f Ce s p ( S ) , so that C T has closed range.

Conversely, suppose that C T has closed range. Then there exists δ 0 such that (27) C T f δ f , for every f Ce s p ( S ) . Choose a positive integer n such that 1 / n < δ . If the set E = { x X : f 0 ( x ) < 1 / n } has positive measure, then, for a given measurable subset F    supp f 0 such that 0 < μ ( F ) < , we have (28) μ T - 1 ( E ) < 1 n μ ( E ) or equivalently (29) C T χ E 1 n χ E . This contradicts inequality (27). Hence, f 0 is bounded away from zero on supp f 0 .

Theorem 8.

Let C T B ( Ce s p ( X ) ) . Then, ker C T * is either zero-dimensional or infinite dimensional.

Proof.

Suppose 0 g ker C T * . Then E = supp g is a set of nonzero measure. Now we can partition E into a sequence { E n } of measurable sets, 0 < μ ( E n ) < . We show that g χ E n T ker C T * . Consider (30) C T * ( g χ E n T ) ( f ) = E ( g χ E n T ) ( C T f ) d μ = E g C T ( χ E n f ) d μ = 0 . Hence, if ker C T * is not zero-dimensional, it is infinite dimensional.

Corollary 9.

Let C T B ( Ce s p ( X ) ) . Then, C T is injective if and only if T is surjective.

Theorem 10.

Let C T B ( Ce s p ( X ) ) . Then, C T has dense range if and only if T - 1 ( s ) = s .

Proof.

Suppose that C T has dense range. Let E s such that χ E Ce s p ( X ) . Then, there exists { f n } Ce s p ( X ) such that C T f n χ E . Now, we can find a subsequence { f n k } of { f n } such that C T f n k χ E a.e. Now, each C T f n k is measurable with respect to T - 1 ( s ) . Therefore, χ E is measurable with respect to T - 1 ( s ) so that χ E = χ T - 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 ) Δ E ) = 0 . Since X is σ -finite, we can find an increasing sequence { F n } of sets of finite measure F n F or T - 1 ( F ) T - 1 ( F n ) 0 . Hence, for given ϵ > 0 , there exists a positive integer n 0 such that μ ( T - 1 ( F F n ) ) < ϵ for every n n 0 . Hence, (31) C T χ F - C T χ F n = C T ( χ F F n ) = χ T - 1 ( F F n ) = ( X ( 1 x 0 x χ T - 1 ( F F n ) d μ ( t ) ) p d μ ( x ) ) p < ϵ , for all n n 0 . Then χ F = χ T - 1 ( F ) ran C T ¯ . This proves that C T has dense range.

Theorem 11.

Let C T B ( Ce s p ( X ) ) . Then, C T is Fredholm if and only if C T is invertible.

Proof.

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

Corollary 12.

Let C T B ( Ce s p ( X ) ) . Then, C T is an isometry if and only if T is measure preserving.

Proof.

If T is measure preserving, then f 0 = 1 a.e. Therefore, (32) C T f = ( X ( 1 x 0 x | f T | ( t ) d μ ( t ) ) p d μ ( x ) ) 1 / p = ( X ( 1 x 0 x f 0 | f | d μ ( t ) ) p d μ ( x ) ) 1 / p = ( X ( 1 x 0 x | f | d μ ( t ) ) p d μ ( x ) ) 1 / p = f . Hence, C T is an isometry. Conversely, if C T is an isometry, then (33) C T χ E = χ E . This implies that μ ( T - 1 ( E ) ) = μ ( E )       for          E s . Hence, f 0 = 1 a.e.

Theorem 13.

Let C T B ( Ce s p ( X ) ) . Then, C T * = P T .

Proof.

Let A s such that 0 < μ ( A ) < . For g Ce s q ( X ) , (34) ( C T * F g ) ( χ A ) = F g ( C T χ A ) = C T χ A g d μ = χ A T g d μ = χ A E ( g ) T - 1 f 0 d μ = F E ( g ) T - 1 f 0 ( χ A ) . Hence, C T * F g = F E ( g ) T - 1 f 0 . After identifying g Ce s q ( X ) with F g ( Ce s p ( X ) ) * , we can write C T * g = E ( g ) T - 1 f 0 = P T g .

Conflict of Interests

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

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