aa r X i v : . [ m a t h . F A ] F e b HYPEREXPANSIVE WEIGHTED COMPOSITION OPERATORS
Y. ESTAREMIAbstract.
In this note unbounded hyperexpansive weighted composition op-erators are investigated. As a consequence unbounded hyperexpansive multi-plication and composition operators are characterized. Introduction and Preliminaries
Weighted composition operators are a general class of operators and they ap-pear naturally in the study of surjective isometries on most of the function spaces,semigroup theory, dynamical systems, Brennans conjecture, etc. This type of op-erators are a generalization of multiplication operators and composition operators.The main subject in the study of composition operators is to describe operatortheoretic properties of C φ in terms of function theoretic properties of φ . The book[3] is a good reference for the theory of composition operators. Weighted composi-tion operators had been studied extensively in past decades. The basic propertiesof weighted composition operators on measurable function spaces are studied byLambert [8, 9], Singh and Manhas [11], Takagi [12], Hudzik and Krbec [7], Cui,Hudzik, Kumar and Maligranda [4], Arora [1], Piotr Budzynski, Zenon Jan Jablon-ski, Il Bong Jung and Jan Stochel [2] and some other mathematicians.In this paper we consider unbounded weighted composition operators on theHilbert space L (Σ) and study hyperexpansive weighted composition operators.As a consequence hyperexpansive multiplication and composition operators arecharacterized.Let H be stand for a Hilbert space and B ( H ) for the Banach algebra of allbounded operators on H . By an operator on H we understand a linear mapping T : D ( T ) ⊆ H → H defined on a linear subspace D ( T ) of H which is called thedomain of T . Set D ∞ ( T ) = ∩ ∞ n =1 D ( T n ). Given an operator T on H , we define thegraph norm k . k T on D ( T ) by k f k T = k f k + k T f k , f ∈ D ( T ) . The next proposition can be easily deduced from the closed graph theorem.
Proposition 1.1. If T is a closed operator on H such that T ( D ( T ) ⊆ ( D ( T ),then T is a bounded operator on the Hilbert space ( D ( T ) , k . k T ).For an operator T on H we set Mathematics Subject Classification.
Key words and phrases. weighted composition operator, hyperexpansive, unbounded operator. Y. ESTAREMI Θ T,n ( f ) = X ≤ i ≤ n ( − i (cid:18) ni (cid:19) k T i ( f ) k , f ∈ D ( T n ) , n ≥ . We recall that an operator T on H is:(i) k -isometry ( k ≥
1) if Θ
T,k ( f ) = 0 for f ∈ D ( T k ),(ii) k -expansive ( k ≥
1) if Θ
T,k ( f ) ≤ f ∈ D ( T k ),(iii) k -hyperexpansive ( k ≥
1) if Θ
T,n ( f ) ≤ f ∈ D ( T n ) and n = 1 , , ..., k .(iv) completely hyperexpansive if Θ T,n ( f ) ≤ f ∈ D ( T n ) and n ≥ Hyperexpansive weighted composition operators
Let ( X, Σ , µ ) be a σ -finite measure space. We denote the collection of (equiva-lence classes modulo sets of zero measure of) Σ-measurable complex-valued func-tions on X by L (Σ) and the support of a function f ∈ L (Σ) is defined as S ( f ) = { x ∈ X ; f ( x ) = 0 } . We also adopt the convention that all comparisonsbetween two functions or two sets are to be interpreted as holding up to a µ -nullset. Denote by L ( µ ) the Hilbert space of all square summable (with respect to µ )Σ-measurable complex functions on X .For each σ -finite subalgebra A of Σ, the conditional expectation, E A ( f ), of f withrespect to A is defined whenever f ≥ f ∈ L . For a sub- σ -finite algebra A ⊆
Σ, the conditional expectation operator associated with A isthe mapping f → E A f , defined for all non-negative f as well as for all f ∈ L (Σ),where E A f , by the Radon-Nikodym theorem, is the unique A -measurable functionsatisfying Z A f dµ = Z A E A f dµ, ∀ A ∈ A . As an operator on L (Σ), E A is an idempotent and E A ( L (Σ)) = L ( A ). If thereis no possibility of confusion we write E ( f ) in place of E A ( f ) [10, 13].For a complex Σ-measurable function u on X . Define the measure µ u : Σ → [0 , ∞ ] by µ u ( E ) = Z E | u | dµ, E ∈ Σ . It is clear that the measure µ u is also σ -finite. By the Radon-Nikodym theorem,if µ u ◦ φ − ≪ µ , then there exists a unique (up to a.e. µ equivalence) Σ-measurablefunction J : X → [0 , ∞ ] such that µ u ( φ − ( E )) = µ u ◦ φ − ( E ) = Z E Jdµ, E ∈ Σ . If µ ◦ φ − ≪ µ , then µ u ◦ φ − ≪ µ . So, by definition of µ u ◦ φ − and applyingconditional expectation with respect to φ − (Σ), we get that J = hE ( | u | ) ◦ φ − ,where h is the Radon-Nykodim derivative dµ ◦ φ − dµ . EIGHTED COMPOSITION OPERATORS 3
Let ( X, Σ , µ ) be a σ -finite measure space, u be a Σ-measurable complex functionand suppose that φ is a mapping from X into X which is measurable (i.e. φ − (Σ) ⊆ Σ). Define the operator uC φ : D ( uC φ ) ⊆ L ( µ ) → L ( µ ) by D ( uC φ ) = { f ∈ L ( µ ) : u.f ◦ φ ∈ L ( µ ) } ,uC φ ( f ) = u.f ◦ φ. Of course such operators may not be well-defined. One can see by direct com-putation that if µ u ◦ φ − ≪ µ , then uC φ is well-defined. And so, if φ is anon-singular transformation, then the operator uC φ is well-defined. Well-definedoperators of the form uC φ ( f ) = u.f ◦ φ acting in L ( µ ) = L ( X, Σ , µ ) with D ( uC φ ) = { f ∈ L ( µ ) : u.f ◦ φ ∈ L ( µ ) } are called weighted composition op-erators. If µ ◦ φ − ≪ µ , then for every f ∈ D ( uC φ ) we have k uC φ ( f ) k = Z X | u | | f ◦ φ | dµ = Z X E ( | u | ) | f | ◦ φdµ = Z X hE ( | u | ) ◦ φ − | f | dµ. By induction we get that for every n ≥ k ( uC φ ) n ( f ) k = Z X | u φ,n | | f ◦ φ n | dµ = Z X J n | f | dµ, for all f ∈ D (( uC φ ) n ). Where u φ,n = u.u ◦ φ.u ◦ φ ...u ◦ φ n − , J n = hE ( J n − | u | ) ◦ φ − , h is the Radon-Nykodim derivative dµ ◦ φ − µ , E is conditional expectation withrespect to φ − (Σ) and J = 1. Lemma 2.1.
Let w = 1 + J and dν = wdµ . Then we have(a) S ( w ) = X and L ( ν ) = D ( uC φ ),(b) And also, the followings are equivalent;(i) uC φ is densely defined.(ii) J < ∞ a.e. µ . Proof. (a) Let f be a measurable function on X . We have k f k µ + k uf ◦ φ k = Z X | f | dµ + Z X | uf ◦ φ | dµ = Z X (1 + J ) | f | dµ = k f k ν . Y. ESTAREMI
This means that, f ∈ D ( uC φ ) if and only if f ∈ L ( ν ). So L ( ν ) = D ( uC φ ).(b) ( i ) → ( ii ) Set F = { J = ∞} . By (a), f | F = 0 a.e. µ for every f ∈ D ( uC φ ).This and (i) implies that f | F = 0 a.e. µ for every f ∈ L ( µ ). So we have χ A ∩ F = 0a.e. µ for all A ∈ Σ with µ ( A ) < ∞ . By the σ -finiteness of µ we have χ F = 0 a.e.i.e µ ( E ) = 0.( ii ) → ( i ) Here we prove that L ( ν ) is dense in L ( µ ). Suppose that f ∈ L ( µ )such that h f, g i = R X f. ¯ gdµ = 0 for all g ∈ L ( ν ). For A ∈ Σ we set A n = { x ∈ A : w ( x ) ≤ n } . It is clear that A n ⊆ A n +1 and X = ∪ ∞ n =1 A n . Since ( X, Σ , µ )is σ -finite, hence X = ∪ ∞ n =1 X n with µ ( X n ) < ∞ . If we set B n = A n ∩ X n , then B n ր A and so f.χ B n ր f.χ A a.e. µ . Since ν ( B n ) ≤ ( n + 1) µ ( B n ) < ∞ , we have χ B n ∈ L ( ν ) and by our assumption R B n f dµ = 0. Therefore by Fatou’s lemma weget that R A f dµ = 0. Thus for all A ∈ Σ we have R A f dµ = 0. This means that f = 0 a.e. µ and so L ( ν ) is dense in L ( µ ).If all functions J i = hE ( J i − | u | ) ◦ φ − , i = 1 , ..., n , are finite valued, where h i is the Radon-Nykodim derivative dµ ◦ φ − i dµ , then we set △ J,n ( x ) = X ≤ i ≤ n ( − i (cid:18) ni (cid:19) J i ( x ) . Proposition 2.2. If D ( uC φ ) is dense in L (Σ), then the following conditionsare equivalent:(i) uC φ ( D ( uC φ )) ⊆ D ( uC φ ).(ii) There exists c > J ≤ c (1 + J ) a.e. µ . Proof. ( i ) → ( ii ). Since uC φ is closed, densely defined and uC φ ( D ( uC φ )) ⊆D ( uC φ ), then by closed graph theorem uC φ is a bounded operator on ( D ( uC φ ) , k . k uC φ ).Hence there exists c > k uC φ ( f ) k uC φ ≤ c k f k uC φ for f ∈ D ( uC φ ). Byreplacing f with uC φ ( f ) we have k ( uC φ ) ( f ) k ≤ k uC φ ( f ) k + k ( uC φ ) ( f ) k ≤ c ( k f k + k uC φ ( f ) k )i.e, Z X J | f | dµ ≤ c ( Z X | f | dµ + Z X J | f | dµ )= Z X c (1 + J ) | f | dµ. This implies that for all f ∈ D ( uC φ ) and also for all f ∈ D ( uC φ ) = L ( µ ) we have Z X ( c (1 + J ) − J ) | f | dµ ≥ EIGHTED COMPOSITION OPERATORS 5 and so J ≤ c (1 + J ) a.e. µ .( ii ) → ( i ). Let f ∈ D ( uC φ ). Then by assumption J ≤ c (1 + J ) a.e. µ , we have Z X | ( uC φ ) ( f ) | dµ = Z X J | f | dµ ≤ c ( Z X | f | dµ + Z X J | f | dµ )= c ( k f k + k uC φ ( f ) k ) < ∞ . Therefore uC φ ( f ) ∈ D ( uC φ ). Remark 2.3. If uC φ ( D ( uC φ )) ⊆ D ( uC φ ) and dν = (1 + J ) dµ , then ( X, Σ , ν )is a σ -finite measure space, ν ◦ φ − is absolutely continuous with respect to ν , L ( ν ) = D ( uC φ ), k . k L ( ν ) is the graph norm of uC φ (considered as an operatorin L ( µ )), and uC φ is a bounded weighted composition operator acting on L ( ν ).Furthermore, if uC φ is k -isometric ( resp. k -expansive, k -hyperexpansive), then sois uC φ as an operator on L ( ν ).If all functions u i and h i for i = 1 , ..., n are finite valued, then we set △ u,n ( x ) = X ≤ i ≤ n ( − i (cid:18) ni (cid:19) u i ( x ) , △ h,n ( x ) = X ≤ i ≤ n ( − i (cid:18) ni (cid:19) h i ( x ) . Corollary 2.4. If D ( C φ ) is dense in L (Σ), then the following conditions areequivalent:(i) C φ ( D ( C φ )) ⊆ D ( C φ ).(ii) There exists c > h ≤ c (1 + h ) a.e. µ . Corollary 2.5. If D ( M u ) is dense in L (Σ), then the following conditions areequivalent:(i) M u ( D ( M u )) ⊆ D ( M u ).(ii) There exists c > u ≤ c (1 + u ) a.e. µ . Proposition 2.6. If D (( uC φ ) n ) is dense in L ( µ ) for a fixed n ≥
1, then:(i) uC φ is k -expansive if and only if △ J,n ( x ) ≤ µ . Y. ESTAREMI (ii) uC φ is k -isometry if and only △ J,n ( x ) = 0 a.e. µ . Proof. (i). Since k ( uC φ ) i ( f ) k = R X J i | f | dµ for all f ∈ D (( uC φ ) i ), we have X ≤ i ≤ n ( − i (cid:18) ni (cid:19) k ( uC φ ) i ( f ) k = X ≤ i ≤ n ( − i (cid:18) ni (cid:19) Z X J i | f | dµ = Z X X ≤ i ≤ n ( − i (cid:18) ni (cid:19) J i | f | dµ = Z X △ J,n ( x ) | f | dµ, for all f ∈ D (( uC φ ) n ). Since ( uC φ ) n is densely defined, then we get that uC φ is k -expansive if and only if △ J,n ( x ) ≤ µ .(ii) Likewise we have uC φ is k -isometry if and only △ J,n ( x ) = 0 a.e. µ . Corollary 2.7 If D (( C φ ) n ) is dense in L ( µ ) for a fixed n ≥
1, then:(i) C φ is k -expansive if and only if △ h,n ( x ) ≤ µ .(ii) C φ is k -isometry if and only △ h,n ( x ) = 0 a.e. µ . Corollary 2.8. If D (( M u ) n ) is dense in L ( µ ) for a fixed n ≥
1, then:(i) M u is k -expansive if and only if △ u,n ( x ) ≤ µ .(ii) M u is k -isometry if and only △ u,n ( x ) = 0 a.e. µ . Proposition 2.9. If D (( uC φ ) ) is dense in L ( µ ) and uC φ is 2-expansive, then:(i) uC φ leaves its domain invariant:(ii) J k ≥ J k − a.e. µ for all k ≥ Proof. (i). By the Proposition 2.3 we get that J ≤ J −
1. Hence for every f ∈ D ( uC φ ) we have k ( uC φ ) ( f ) k = Z X J | f | dµ ≤ Z X J | f | dµ − Z X | f | dµ < ∞ , so uC φ ( f ) ∈ D ( uC φ ). EIGHTED COMPOSITION OPERATORS 7 (ii) Since uC φ leaves its domain invariant, then D ( uC φ ) ⊆ D ∞ ( uC φ ). So bylemma 3.2 (iii) of [6] we get that k ( uC φ ) k ( f ) k ≥ k ( uC φ ) k − ( f ) k for all f ∈D ( uC φ ) and k ≥ Z X ( J k − J k − ) | f | dµ ≥ , f ∈ D ( uC φ ) , so this leads to J k ≥ J k − a.e. µ . Corollary 2.10. If D (( C φ ) ) is dense in L ( µ ) and C φ is 2-expansive, then:(i) C φ leaves its domain invariant:(ii) h k ≥ h k − a.e. µ for all k ≥ Corollary 2.11. If D (( M u ) ) is dense in L ( µ ) and M u is 2-expansive, then:(i) M u leaves its domain invariant:(ii) u k ≥ u k − a.e. µ for all k ≥ ϕ on N is said to be completely alternating if P ≤ i ≤ n ( − i (cid:18) ni (cid:19) ϕ ( m + i ) ≤ m ≥ n ≥
1. The next theorem is adirect consequence of proposition 2.3 and 2.4.
Theorem 2.12. If D (( uC φ ) ) is dense in L ( µ ) and k ≥ uC φ is k -hyperexpansive if and only if △ J,i ( x ) ≤ µ for i = 1 , ..., k .(ii) uC φ is completely hyperexpansive if and only if { J i } ∞ i =0 is a completely al-ternating sequence for almost every x ∈ X . Corollary 2.13. If D (( C φ ) ) is dense in L ( µ ) and k ≥ C φ is k -hyperexpansive if and only if △ h,i ( x ) ≤ µ for i = 1 , ..., k .(ii) C φ is completely hyperexpansive if and only if { h i } ∞ i =0 is a completely alter-nating sequence for almost every x ∈ X . Corollary 2.14. If D (( M u ) ) is dense in L ( µ ) and k ≥ M u is k -hyperexpansive if and only if △ u,i ( x ) ≤ µ for i = 1 , ..., k . Y. ESTAREMI (ii) M u is completely hyperexpansive if and only if { u i } ∞ i =0 is a completely al-ternating sequence for almost every x ∈ X .Notice that in the same way we can characterize alternatingly hyperexpansiveweighted composition operators.We say that the σ -algebra φ − (Σ) is essentially all of Σ with respect to µ u if andonly of given A ∈ Σ there is B ∈ Σ with the symmetric difference φ − ( B ) △ A =( φ − ( B ) \ A ) ∪ ( A \ φ − ( B )) having µ u ( φ − ( B ) △ A ) = 0.The following proposition characterizes 2-expansive weighted composition oper-ators on the measure space ( X, Σ , µ ) such that µ u ( X ) < ∞ . Theorem 2.15.
Let uC φ be 2-expansive operator.(i) Let ( X, Σ , µ ) is an infinite measure space such that µ u ( X ) < ∞ and D (( uC φ ) )is dense in L ( µ ).(ii) Let ( X, Σ , µ ) is a measure space such that µ u ( X ) < ∞ , u ≤ µ and D (( uC φ ) ) is dense in L ( µ ).If the conditions (i) or (ii) holds, then uC φ is an isometry.(iii) If uC φ is densely defined, u = 0 a.e. µ and the sigma algebra φ − (Σ) isessentially all of Σ, with respect to µ , then uC φ is a unitary operator. Proof. (i) It follows from Proposition 2.4 that uC φ leaves its domain invariantand J ≥ µ . Suppose that (a) holds and contrary to our claim, there exists B ⊆ X such that µ ( B ) > J ≥ ǫ + 1 on B for some ǫ >
0. Then we have ∞ > µ u ( X ) = µ u ( φ − ( X \ B )) + µ u ( φ − ( B )) ≥ (1 + ǫ ) µ ( B ) + µ ( X \ B ) > µ ( X ) , which is a contradiction. Thus J = 1 a.e. µ .If (ii) holds, then by the same method we conclude that µ ( X ) > µ ( X ) which isa contradiction. These imply that uC φ is an isometry.(iii) Let u = 0 a.e. µ and the sigma algebra φ − (Σ) be essentially all of Σ, withrespect to µ . This implies that uC φ is dense range. Then by [[6], proposition 3.5]we get that uC φ is unitary. Corollary 2.16.
Let C φ be 2-expansive operator.(i) If ( X, Σ , µ ) is a finite measure space and D (( C φ ) ) is dense in L ( µ ), then C φ is an isometry. EIGHTED COMPOSITION OPERATORS 9 (ii) If C φ is densely defined and the sigma algebra φ − (Σ) is essentially all of Σ,then C φ is a unitary operator. Corollary 2.17.
Let M u be 2-expansive operator.(i) Let ( X, Σ , µ ) is an infinite measure space such that µ u ( X ) < ∞ and D (( M u ) )is dense in L ( µ ).(ii) Let ( X, Σ , µ ) is a measure space such that µ u ( X ) < ∞ , u ≤ µ and D (( M u ) ) is dense in L ( µ ).If the conditions (i) or (ii) holds, then M u is an isometry.(iii) If M u is densely defined, u = 0 a.e. µ , then M u is a unitary operator.Now, let m = { m n } ∞ n =1 be a sequence of positive real numbers. Consider thespace l ( m ) = L ( N , N , µ ), where 2 N is the power set of natural numbers and µ is a measure on 2 N defined by µ ( { n } ) = m n . Let u = { u n } ∞ n =1 be a sequence ofcomplex numbers. Let ϕ : N → N be a non-singular measurable transformation;i.e. µ ◦ ϕ − ≪ µ . Direct computation shows that h ( k ) = 1 m k X j ∈ ϕ − ( k ) m j , E ϕ ( f )( k ) = P j ∈ ϕ − ( ϕ ( k )) f j m j P j ∈ ϕ − ( ϕ ( k )) m j , for all non-negative sequence f = { f n } ∞ n =1 and k ∈ N . So J ( k ) = 1 m k X j ∈ ϕ − ( k ) | u j | m j . This observations lets us to consider the weighted composition operators on discretemeasure space ( N , µ, Σ). If uC φ is a weighted composition operator on l ( m ), then D ( uC φ ) = { f = { f n } ∈ l ( m ) : ∞ X n =0 ( X j ∈ ϕ − ( n ) | u j | m j ) | f n | < ∞}k uf ◦ φ k = Z X | uf ◦ φ | dµ = ∞ X n =0 J ( n ) m n | f n | = ∞ X n =0 ( X j ∈ ϕ − ( n ) | u j | m j ) | f n | . References [1] S. C. Arora, Gopal Datt and Satish Verma, Composition operators on Lorentz spaces, Bull.Austral. Math. Soc. (2007), 205-214.[2] Piotr Budzynski, Zenon Jan Jablonski, Il Bong Jung, Jan Stochel, Unbounded WeightedComposition Operators in L2-Spaces, arXiv:1310.3542. Y. ESTAREMI [3] C. C. Cowen and B. D. MacCluer, Composition Operators on Spaces of Analytic Functions,Studies in Advanced Mathematics, CRC Press, Boca Raton, Fla, USA, (1995).[4] Y. Cui, H. Hudzik, R. Kumar and L. Maligranda, Composition operators in Orlicz spaces, J.Aust. Math. Soc. (2004), 189-206.[5] J. Jablonski, Hyperexpansive composition operators, Math. Proc. Camb. Phil. Soc. (2003), 513-526.[6] J. Jablonski and J. Stochel, Unbounded 2-hyperexpansive operators, Proc. Edinburgh Math.Soc. (2001), 613-629.[7] H. Hudzik and M. Krbec, On non-effective weights in Orlicz spaces. Indag. Math. (N.S.) (2007), 215-231.[8] A. Lambert, Localising sets for sigma-algebras and related point transformations, Proc. Roy.Soc. Edinburgh Sect. A (1991), 111-118.[9] A. Lambert, Operator algebras related to measure preserving transformations of finite order,Rocky Mountain J. Math. (1984), 341-349.[10] M. M. Rao, Conditional measure and applications, Marcel Dekker, New York, 1993.[11] R. K. Singh and J. S. Manhas, Composition operators on function spaces, North HollandMath. Studies 179, Amsterdam 1993.[12] H. Takagi, Compact weighted composition operators on L p , Proc. Amer. Math. Soc. (1992), 505-511.[13] A. C. Zaanen, Integration, 2nd ed., North-Holland, Amsterdam, 1967. y. estaremi E-mail address : [email protected]@gmail.com