Differences of composition operators from analytic Besov spaces into little Bloch type spaces
aa r X i v : . [ m a t h . F A ] M a y DIFFERENCES OF COMPOSITION OPERATORS FROMANALYTIC BESOV SPACES INTO LITTLE BLOCH TYPESPACES
AJAY K. SHARMA AND SEI-ICHIRO UEKI
Abstract.
The purpose of this paper is to describe the characterization forthe compact difference of two composition operators acting between analyticBesov spaces and the weighted little Bloch type space over the unit disk. Introduction
Let D denote the open unit disk in the complex plane C and dA the normalizedarea measure on D . Let H ( D ) be the set of all analytic functions on D . When1 < p < ∞ , a function f ∈ H ( D ) is said to be in the analytic Besov space B p ifand only if Z D | f ′ ( z ) | p (1 − | z | ) p − dA ( z ) < ∞ . And a norm k · k p on B p is defined by k f k p = | f (0) | + (cid:20)Z D | f ′ ( z ) | p (1 − | z | ) p − dA ( z ) (cid:21) /p . For the case p = 1, the above integrable condition is satisfied by only constantfunctions. Thus the definition of the space B is complicated and there are severalways to define B . If 1 < p < ∞ , it is well known that f ∈ B p is equivalent to Z M pp ( r, f ′′ )(1 − r ) p − dr < ∞ . In the case p = 1, the above condition becomes R M ( r, f ′′ ) dr < ∞ . Hence we candefine the space B by the condition Z D | f ′′ ( z ) | dA ( z ) < ∞ . For w ∈ D , let α w ( z ) be the conformal automorphism of D defined by α w ( z ) =( w − z ) / (1 − wz ) ( z ∈ D ). Each function f ∈ B has an atomic decomposition, thatis there exist sequences { c j } ∈ l and { w j } ⊂ D such that f ( z ) = c + ∞ X j =1 c j α w j ( z ) ( z ∈ D ) . Mathematics Subject Classification.
Primary 30H25; Secondary 30H30.
Key words and phrases.
Besov spaces, little Bloch spaces, composition operators.
By using this representation, a norm k · k on B is defined by k f k = inf ∞ X j =0 | c j | , where the infimum is taken over all { c j } ∈ l satisfy the above atomic decompositionfor f ∈ B . It is known that k f k is comparable to | f (0) | + | f ′ (0) | + Z D | f ′′ ( z ) | dA ( z ) . For more details about analytic Besov spaces, we can refer to monographs [12, 13].Next we will introduce the weighted Bloch type space. Throughout this paper,let ν be a positive continuous radial function on D . Here “radial” means that ν ( z ) = ν ( | z | ) for z ∈ D . The weighted Bloch type space B ν is the space of all f ∈ H ( D ) which satisfy sup z ∈ D ν ( z ) | f ′ ( z ) | < ∞ , and the little Bloch type space B ν, consists of all f ∈ B ν satisfying ν ( z ) | f ′ ( z ) | → | z | → − . It is easy to seethat the space B ν, is a closed subspace in B ν . These Bloch type spaces have beenappeared in studies on composition or integral operators. For instance, S. Stevi´cand his collaborators have many studies about these operators acting on Bloch typespaces; see [9, 10, 11] and the related references therein.One of the major subjects in the field of analytic function spaces and operatortheory is studies on composition operators. For an analytic self-map ϕ of D , thecomposition operator C ϕ is defined by C ϕ f = f ◦ ϕ ( f ∈ H ( D )). This compositionoperator has been studied extensively on various analytic function spaces. The aimof these studies is to explore the relation between operator-theoretic behaviors of C ϕ and function-theoretic properties of the map ϕ . Over the past few decades, a con-siderable number of studies have been conducted on the difference of compositionoperators on analytic function spaces. Shapiro and Sundberg [7] and MacCluer etal. [2] studied a compact difference of composition operators on the Hardy spacesand topological structures of the space of composition operators. Moorhouse [4]and Saukko [5, 6] have investigated characterizations for the compactness of thesame operator on the weighted Bergman spaces, Hosokawa and Ohno [1] have con-sidered it acting on the Bloch spaces. They used the pseudo-hyperbolic metric togive equivalent conditions for the compactness of the difference operator of compo-sitions.Recently, motivated by these results, we have investigated this type operatorfrom the analytic Besov space B p into the Bloch type space B ν in [8]. In that paper,we dealt with the case C ϕ − C ψ : B p → B ν only. Hence the purpose of this paperis to describe equivalent conditions for the compactness of C ϕ − C ψ : B p → B ν, .When we consider the case that the range of C ϕ − C ψ is different from the domainof it, we have to take notice of the boundedness of it because a pair { ϕ, ψ } does notalways induce the bounded difference operator of compositions. In Section 3, wewill give characterizations for the boundedness of C ϕ − C ψ which the range is B ν, .By applying this result for the boundedness, we will describe characterizations forthe compactness of C ϕ − C ψ . Section 4 is devoted to explain the details of them.Throughout this paper, the notation A . B means that there exists a positiveconstant C such that A ≤ CB . Of course, the constant C is independent of afunction f , a point z ∈ D and related parameters { t, r } . Moreover, if both A . B and B . A hold, then one says that A ≈ B . IFFERENCES OF COMPOSITION OPERATORS 3 Preliminaries
We will need the following results in Section 3 and 4.
Lemma 1.
Let ≤ p < ∞ and f ∈ B p . Then | f ′ ( z ) | . k f k p − | z | for all z ∈ D .Proof. We have to consider the two cases p = 1 and p = 1. For the case p = 1,by the definition of the space B p , f ∈ B p if and only if f ′ belongs to the classicalweighted Bergman space L pa ( dA p − ). Hence f ′ has the following point evaluationestimate: | f ′ ( z ) | ≤ k f ′ k L pa ( dA p − ) − | z | for all z ∈ D . Since k f ′ k L pa ( dA p − ) ≤ k f k p , we obtain the desired estimate. Toprove the case p = 1, we use the atomic decomposition of f ∈ B . If f ∈ B , wecan choose sequences { c j } ∈ l and { w j } ⊂ D such that f = c + P c j α w j . Thuswe have | f ( z ) | . P | c j | for all z ∈ D . By taking the infimum with respect to allsuch representation of f , we obtain | f ( z ) | . k f k for all z ∈ D . An application ofCauchy’s estimate to f ′ on the circle with center at z and radius (1 − | z | ) / | f ′ ( z ) | . k f k / (1 − | z | ) for all z ∈ D . (cid:3) Lemma 2.
Let ≤ p < ∞ and f ∈ B p . Then | (1 − | z | ) f ′ ( z ) − (1 − | w | ) f ′ ( w ) | . k f k p ρ ( z, w ) for all { z, w } ⊂ D .Proof. In [1, Proposition 2.2], Hosokawa and Ohno proved that | (1 − | z | ) f ′ ( z ) − (1 − | w | ) f ′ ( w ) | . ρ ( z, w ) sup ζ ∈ D (1 − | ζ | ) | f ′ ( ζ ) | for f belongs to the Bloch space B and { z, w } ⊂ D . Since Lemma 1 imply that B ⊂ B p (1 ≤ p < ∞ ) and sup ζ ∈ D (1 − | ζ | ) | f ′ ( ζ ) | . k f k p , the desired estimate canbe verified by the above estimate. (cid:3) A compact subset of B ν, can be characterized as following. The same result forthe usual little Bloch space B was proved by Madigan and Matheson [3]. By aslightly modification of their proof, we can prove the following lemma. Lemma 3.
A closed subset L in B ν, is compact if and only if it is a bounded subsetin B ν and satisfies lim | z |→ − sup f ∈ L ν ( z ) | f ′ ( z ) | = 0 . The following result is appeared in our previous work [8]. We will need it in theargument of the compactness in Section 4.
Theorem 1.
Let ≤ p < ∞ and { ϕ, ψ } a pair of analytic self-maps of D . Thenthe following statements are equivalent: (i) C ϕ − C ψ : B p → B ν is bounded, A.K. SHARMA AND S. UEKI (ii) ϕ and ψ satisfy the following two conditions: sup z ∈ D ν ( z ) | ϕ ′ ( z ) | − | ϕ ( z ) | ρ ( ϕ ( z ) , ψ ( z )) < ∞ and sup z ∈ D (cid:12)(cid:12)(cid:12)(cid:12) ν ( z ) ϕ ′ ( z )1 − | ϕ ( z ) | − ν ( z ) ψ ′ ( z )1 − | ψ ( z ) | (cid:12)(cid:12)(cid:12)(cid:12) < ∞ , (iii) ϕ and ψ satisfy the following two conditions: sup z ∈ D ν ( z ) | ψ ′ ( z ) | − | ψ ( z ) | ρ ( ϕ ( z ) , ψ ( z )) < ∞ and sup z ∈ D (cid:12)(cid:12)(cid:12)(cid:12) ν ( z ) ϕ ′ ( z )1 − | ϕ ( z ) | − ν ( z ) ψ ′ ( z )1 − | ψ ( z ) | (cid:12)(cid:12)(cid:12)(cid:12) < ∞ . Boundedness of C ϕ − C ψ Before considering the compactness of C ϕ − C ψ , we have to mention the bound-edness of it. The following Theorem 2 can be found in [1, Theorem 3.4]. Theyproved the result for the case that C ϕ − C ψ is acting on the little Bloch space B . Under the assumption on the boundedness of C ϕ − C ψ and the density of thepolynomial set in the domain space, we can generalize their result as following. Theorem 2.
Let X be a Banach space of analytic functions over D which thepolynomial set is dense in X . For each pair { ϕ, ψ } of analytic self-maps of D with C ϕ − C ψ : X → B ν is bounded, the following conditions are equivalent: (a) C ϕ − C ψ : X → B ν, is bounded, (b) ϕ − ψ ∈ B ν, and ϕ − ψ ∈ B ν, , (c) ϕ − ψ ∈ B ν, and lim | z |→ − ν ( z ) | ϕ ( z ) − ψ ( z ) | max {| ϕ ′ ( z ) | , | ψ ′ ( z ) |} = 0 . Proof.
The direction (a) ⇒ (b) is verified by test functions p ( z ) = z and p ( z ) = z easily. Hence it is enough to prove directions (b) ⇒ (c) and (c) ⇒ (a). Now we willprove (b) ⇒ (c). Since ϕ − ψ ∈ B ν, implies ν ( z ) | ϕ ′ ( z ) − ψ ′ ( z ) | → | z | → − and ϕ − ψ ∈ B ν, implies ν ( z ) | ϕ ( z ) ϕ ′ ( z ) − ψ ( z ) ψ ′ ( z ) | → | z | → − , we obtainthat ν ( z ) | ϕ ( z ) − ψ ( z ) || ϕ ′ ( z ) |≤ ν ( z ) | ϕ ( z ) ϕ ′ ( z ) − ψ ( z ) ψ ′ ( z ) | + ν ( z ) | ϕ ′ ( z ) − ψ ′ ( z ) || ψ ( z ) |≤ ν ( z ) | ϕ ( z ) ϕ ′ ( z ) − ψ ( z ) ψ ′ ( z ) | + ν ( z ) | ϕ ′ ( z ) − ψ ′ ( z ) | → , as | z | → − . We also have ν ( z ) | ϕ ( z ) − ψ ( z ) || ψ ′ ( z ) | → | z | → − , and so thecondition (c) is true. In order to prove (c) ⇒ (a), we assume (c). For each n ≥ p n ( z ) = z n . Then( C ϕ − C ψ ) p n ( z ) = ϕ n ( z ) − ψ n ( z ) = ( ϕ ( z ) − ψ ( z )) n − X k =0 ϕ n − − k ( z ) ψ k ( z ) . IFFERENCES OF COMPOSITION OPERATORS 5
We will claim that ( C ϕ − C ψ ) p n ∈ B ν, . Since n − X k =0 ϕ n − − k ( z ) ψ k ( z ) ! ′ = ( n − ϕ n − ( z ) ϕ ′ ( z ) + ( n − ϕ n − ( z ) ϕ ′ ( z ) ψ ( z ) + ϕ n − ( z ) ψ ′ ( z )+ · · · + ϕ ′ ( z ) ψ n − ( z ) + ( n − ϕ ( z ) ψ n − ( z ) ψ ′ ( z ) + ( n − ψ n − ( z ) ψ ′ ( z ) , we have that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n − X k =0 ϕ n − − k ( z ) ψ k ( z ) ! ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ n ( n − | ϕ ′ ( z ) | + | ψ ′ ( z ) | ) . Hence this inequality gives that ν ( z ) | (( C ϕ − C ψ ) p n ) ′ ( z ) |≤ nν ( z ) | ϕ ′ ( z ) − ψ ′ ( z ) | + n ( n − ν ( z ) | ϕ ( z ) − ψ ( z ) | ( | ϕ ′ ( z ) | + | ψ ′ ( z ) | ) . Combining this estimate with the condition (c), we see that ( C ϕ − C ψ ) p n ∈ B ν, ,and so ( C ϕ − C ψ ) p ∈ B ν, for all analytic polynomial p . Since the polynomial setis dense in X, C ϕ − C ψ : X → B ν is bounded and B ν, is closed in B ν , we also see( C ϕ − C ψ ) f ∈ B ν, for f ∈ X . This implies the boundedness of C ϕ − C ψ : X →B ν, . (cid:3) Lemma 4.
For ≤ p < ∞ , the polynomial set is dense in B p .Proof. Each dilated function f r is analytic in the closed unit disk D , and so itbelongs to B p . Since f r is approximated by polynomials in B p , it is enough toprove that every f ∈ B p satisfies k f − f r k p → r → − . First we will considerthe case p >
1. Fix ε >
0. Since f ∈ B p , there exists an R ∈ (0 ,
1) such that R D \ R D | f ′ ( z ) | p (1 − | z | ) p − dA ( z ) < ε . Noting that | f ′ | p is subharmonic in D , wehave Z D \ R D | f ′ ( rz ) | p (1 − | z | ) p − dA ( z ) = 2 Z R t (1 − t ) p − dt Z π | f ′ ( rte iθ ) | p dθ π ≤ Z R t (1 − t ) p − dt Z π | f ′ ( te iθ ) | p dθ π = Z D \ R D | f ′ ( z ) | p (1 − | z | ) p − dA ( z ) < ε for any r ∈ (0 , k f − f r k pp = Z D | f ′ ( z ) − rf ′ ( rz ) | p (1 − | z | ) p − dA ( z ) . (1 − r ) p Z D | f ′ ( rz ) | p (1 − | z | ) p − dA ( z )+ Z D | f ′ ( z ) − f ′ ( rz ) | p (1 − | z | ) p − dA ( z ) . (1 − r ) p k f k pp + ε + Z R D | f ′ ( z ) − f ′ ( rz ) | p (1 − | z | ) p − dA ( z ) . A.K. SHARMA AND S. UEKI
Since f ′ is uniform continuous on R D , it follows from this estimate that k f − f r k p → r → − . For the case p = 1, we obtain k f − f r k . Z D | f ′′ ( z ) − r f ′′ ( rz ) | dA ( z ) ≤ (1 − r ) Z D | f ′′ ( rz ) | dA ( z ) + Z D | f ′′ ( rz ) − f ′′ ( z ) | dA ( z ) . By the same argument as in the case p >
1, these inequalities also show that k f − f r k → r → − . (cid:3) Corollary 1.
Let ≤ p < ∞ and { ϕ, ψ } a pair of analytic self-maps of D whichinduces the bounded operator C ϕ − C ψ : B p → B ν . Then the following conditionsare equivalent: (a) C ϕ − C ψ : B p → B ν, is bounded, (b) ϕ − ψ ∈ B ν, and ϕ − ψ ∈ B ν, , (c) ϕ − ψ ∈ B ν, and lim | z |→ − ν ( z ) | ϕ ( z ) − ψ ( z ) | max {| ϕ ′ ( z ) | , | ψ ′ ( z ) |} = 0 . Compactness of C ϕ − C ψ Theorem 3.
Let ≤ p < ∞ . For each { ϕ, ψ } a pair of analytic self-maps of D , C ϕ − C ψ : B p → B ν, is compact if and only if ϕ and ψ satisfy the following twocondtions: (a) lim | z |→ − max (cid:26) | ϕ ′ ( z ) | − | ϕ ( z ) | , | ψ ′ ( z ) | − | ψ ( z ) | (cid:27) ν ( z ) ρ ( ϕ ( z ) , ψ ( z )) = 0 , (b) lim | z |→ − (cid:12)(cid:12)(cid:12)(cid:12) ν ( z ) ϕ ′ ( z )1 − | ϕ ( z ) | − ν ( z ) ψ ′ ( z )1 − | ψ ( z ) | (cid:12)(cid:12)(cid:12)(cid:12) = 0 .Proof. Now we assume that conditions (a) and (b) are true. Let K = { f ∈ B p : k f k p ≤ } closed unit ball in B p . In order to prove the compactness of C ϕ − C ψ : B p → B ν, , by Lemma 3, we may prove that C ϕ − C ψ : B p → B ν, is bounded andlim | z |→ sup f ∈ K ν ( z ) | (( C ϕ − C ψ ) f ) ′ ( z ) | = 0 . (1)By Theorem 1 we see that (a) and (b) imply the boundedness of C ϕ − C ψ : B p →B ν . Thus we will claim that ϕ and ψ satisfy the condition (c) in Corollary 1. Since | ϕ ( z ) − ψ ( z ) | ≤ ρ ( ϕ ( z ) , ψ ( z )) for z ∈ D , we see that ν ( z ) | ϕ ( z ) − ψ ( z ) || ϕ ′ ( z ) | ≤ ν ( z ) | ϕ ′ ( z ) | − | ϕ ( z ) | ρ ( ϕ ( z ) , ψ ( z )) , and ν ( z ) | ϕ ( z ) − ψ ( z ) || ψ ′ ( z ) | ≤ ν ( z ) | ψ ′ ( z ) | − | ψ ( z ) | ρ ( ϕ ( z ) , ψ ( z )) , and so the condition (a) showslim | z |→ ν ( z ) | ϕ ( z ) − ψ ( z ) | max {| ϕ ′ ( z ) | , | ψ ′ ( z ) |} = 0 . Moreover we also obtain that ν ( z ) | ϕ ′ ( z ) − ψ ′ ( z ) | = ν ( z ) (cid:12)(cid:12)(cid:12)(cid:12) (1 − | ϕ ( z ) | ) ϕ ′ ( z )1 − | ϕ ( z ) | − (1 − | ψ ( z ) | ) ψ ′ ( z )1 − | ψ ( z ) | (cid:12)(cid:12)(cid:12)(cid:12) IFFERENCES OF COMPOSITION OPERATORS 7 = (cid:12)(cid:12)(cid:12)(cid:12) ν ( z ) ϕ ′ ( z )1 − | ϕ ( z ) | − ν ( z ) ψ ′ ( z )1 − | ψ ( z ) | (cid:12)(cid:12)(cid:12)(cid:12) + ν ( z ) (cid:12)(cid:12)(cid:12)(cid:12) − | ϕ ( z ) | ϕ ′ ( z )1 − | ϕ ( z ) | + | ψ ( z ) | ϕ ′ ( z )1 − | ϕ ( z ) | − | ψ ( z ) | ϕ ′ ( z )1 − | ϕ ( z ) | + | ψ ( z ) | ψ ′ ( z )1 − | ψ ( z ) | (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) ν ( z ) ϕ ′ ( z )1 − | ϕ ( z ) | − ν ( z ) ψ ′ ( z )1 − | ψ ( z ) | (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12) | ϕ ( z ) | − | ψ ( z ) | (cid:12)(cid:12) ν ( z ) | ϕ ′ ( z ) | − | ϕ ( z ) | ≤ (cid:12)(cid:12)(cid:12)(cid:12) ν ( z ) ϕ ′ ( z )1 − | ϕ ( z ) | − ν ( z ) ψ ′ ( z )1 − | ψ ( z ) | (cid:12)(cid:12)(cid:12)(cid:12) + 4 ρ ( ϕ ( z ) , ψ ( z )) ν ( z ) | ϕ ′ ( z ) | − | ϕ ( z ) | . Hence (a) and (b) show that ν ( z ) | ϕ ′ ( z ) − ψ ′ ( z ) | → | z | → − , that is ϕ − ψ ∈ B ν, .By Corollary 1, we see that C ϕ − C ψ : B p → B ν, is bounded.Next we will prove the equation (1). Fix z ∈ D and f ∈ K . Thus we have ν ( z ) | (( C ϕ − C ψ ) f ) ′ ( z ) | = ν ( z ) | f ′ ( ϕ ( z )) ϕ ′ ( z ) − f ′ ( ψ ( z )) ψ ′ ( z ) | = ν ( z ) (cid:12)(cid:12)(cid:12)(cid:12) ϕ ′ ( z )1 − | ϕ ( z ) | (1 − | ϕ ( z ) | ) f ′ ( ϕ ( z )) − ψ ′ ( z )1 − | ψ ( z ) | (1 − | ψ ( z ) | ) f ′ ( ψ ( z )) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) ν ( z ) ϕ ′ ( z )1 − | ϕ ( z ) | − ν ( z ) ψ ′ ( z )1 − | ψ ( z ) | (cid:12)(cid:12)(cid:12)(cid:12) (1 − | ϕ ( z ) | ) | f ′ ( ϕ ( z )) | + (cid:12)(cid:12) (1 − | ϕ ( z ) | ) f ′ ( ϕ ( z )) − (1 − | ψ ( z ) | ) f ′ ( ψ ( z )) (cid:12)(cid:12) ν ( z ) | ψ ′ ( z ) | − | ψ ( z ) | . Combining this with Lemma 1 and 2, we obtain ν ( z ) | (( C ϕ − C ψ ) f ) ′ ( z ) | . (cid:12)(cid:12)(cid:12)(cid:12) ν ( z ) ϕ ′ ( z )1 − | ϕ ( z ) | − ν ( z ) ψ ′ ( z )1 − | ψ ( z ) | (cid:12)(cid:12)(cid:12)(cid:12) + ν ( z ) | ψ ′ ( z ) | − | ψ ( z ) | ρ ( ϕ ( z ) , ψ ( z ))for any z ∈ D and f ∈ K . Conditions (a) and (b) imply (1). By Lemma 3 we seethat ( C ϕ − C ψ )( K ) is a compact subset in B ν, . Hence C ϕ − C ψ : B p → B ν, iscompact.To prove that the compactness of C ϕ − C ψ gives conditions (a) and (b), we takea sequence { z n } of D with | z n | → − as n → ∞ arbitrary. Put f n ( z ) = ϕ ( z n ) − z − ϕ ( z n ) z , and g n ( z ) = ϕ ( z n ) − z − ϕ ( z n ) z ! for n ≥ z ∈ D . Then { f n , g n } ⊂ B p and we can choose a positive constant C which is independent of n , ϕ and ψ such that k f n k p ≤ C and k g n k p ≤ C . Let K C = { f ∈ B p : k f k p ≤ C } . By Lemma 3, the compactness of C ϕ − C ψ implieslim n → sup f ∈ K C ν ( z n ) | (( C ϕ − C ψ ) f ) ′ ( z n ) | = 0 . (2)By the definition of f n , we have ν ( z n ) | (( C ϕ − C ψ ) f n ) ′ ( z n ) | = ν ( z n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − ϕ ′ ( z n )1 − | ϕ ( z n ) | + ψ ′ ( z n )(1 − | ϕ ( z n ) | )(1 − ϕ ( z n ) ψ ( z n )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ (cid:12)(cid:12)(cid:12)(cid:12) ν ( z n ) | ϕ ′ ( z n ) | − | ϕ ( z n ) | − ν ( z n ) | ψ ′ ( z n ) | − | ψ ( z n ) | (1 − ρ ( ϕ ( z n ) , ψ ( z n ))) ) (cid:12)(cid:12)(cid:12)(cid:12) . (3) A.K. SHARMA AND S. UEKI
Hence (2) giveslim n →∞ (cid:12)(cid:12)(cid:12)(cid:12) ν ( z n ) | ϕ ′ ( z n ) | − | ϕ ( z n ) | − ν ( z n ) | ψ ′ ( z n ) | − | ψ ( z n ) | (1 − ρ ( ϕ ( z n ) , ψ ( z n ))) ) (cid:12)(cid:12)(cid:12)(cid:12) = 0 . (4)Since g ′ n ( ϕ ( z n )) = 0, we have ν ( z n ) | (( C ϕ − C ψ ) g n ) ′ ( z n ) | = 2 ν ( z n ) | ϕ ( z n ) − ψ ( z n ) || ψ ′ ( z n ) || − ϕ ( z n ) ψ ( z n ) | (1 − | ϕ ( z n ) | )= 2 ν ( z n ) | ψ ′ ( z n ) | − | ψ ( z n ) | ρ ( ϕ ( z n ) , ψ ( z n ))(1 − ρ ( ϕ ( z n ) , ψ ( z n )) ) . The equation (2) also giveslim n →∞ ν ( z n ) | ψ ′ ( z n ) | − | ψ ( z n ) | ρ ( ϕ ( z n ) , ψ ( z n ))(1 − ρ ( ϕ ( z n ) , ψ ( z n )) ) = 0 . (5)By replacing the role of ϕ and ψ in definitions f n and g n , we also see that ϕ and ψ satisfy lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12) ν ( z n ) | ψ ′ ( z n ) | − | ψ ( z n ) | − ν ( z n ) | ϕ ′ ( z n ) | − | ϕ ( z n ) | (1 − ρ ( ϕ ( z n ) , ψ ( z n ))) ) (cid:12)(cid:12)(cid:12)(cid:12) = 0 , (6)and lim n →∞ ν ( z n ) | ϕ ′ ( z n ) | − | ϕ ( z n ) | ρ ( ϕ ( z n ) , ψ ( z n ))(1 − ρ ( ϕ ( z n ) , ψ ( z n )) ) = 0 . (7)Now we assume that lim n →∞ ν ( z n ) | ϕ ′ ( z n ) | − | ϕ ( z n ) | ρ ( ϕ ( z n ) , ψ ( z n )) = 0 . Then (7) indicates that 1 − ρ ( ϕ ( z n ) , ψ ( z n )) → n → ∞ . By (4) we obtainlim n →∞ ν ( z n ) | ϕ ′ ( z n ) | − | ϕ ( z n ) | = 0 . Since ρ ( ϕ ( z n ) , ψ ( z n )) <
1, this claim contradicts our assumption. By the sameargument with (5) and (6), we also have thatlim n →∞ ν ( z n ) | ψ ′ ( z n ) | − | ψ ( z n ) | ρ ( ϕ ( z n ) , ψ ( z n )) = 0 . (8)Since { z n } ⊂ D with | z n | → n → ∞ was arbitrary, these imply the condition(a) holds. Furthemore, the estimate (3) gives ν ( z n ) | (( C ϕ − C ψ ) f n ) ′ ( z n ) |≥ (cid:12)(cid:12)(cid:12)(cid:12) ν ( z n ) | ϕ ′ ( z n ) | − | ϕ ( z n ) | − ν ( z n ) | ψ ′ ( z n ) | − | ψ ( z n ) | (cid:12)(cid:12)(cid:12)(cid:12) − ν ( z n ) | ψ ′ ( z n ) | − | ψ ( z n ) | ρ ( ϕ ( z n ) , ψ ( z n )) . By (2) and (8), we obtainlim n →∞ (cid:12)(cid:12)(cid:12)(cid:12) ν ( z n ) | ϕ ′ ( z n ) | − | ϕ ( z n ) | − ν ( z n ) | ψ ′ ( z n ) | − | ψ ( z n ) | (cid:12)(cid:12)(cid:12)(cid:12) = 0 , and so this indicates the condition (b). (cid:3) IFFERENCES OF COMPOSITION OPERATORS 9
Acknowledgement.
This research is partly supported by JSPS KAKENHI Grants-in-Aid for Scientific Research (C), Grant Number 17K05282. Partial work on thispaper was done while the second author visited the Department of Mathematics,Central University of Jammu, Jammu. He wishes to thank Central University ofJammu for hosting his visit. The first author is thankful to NBHM(DAE)(India)for the project grant No. 02011/30/2017/R&D II/12565.
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Ajay K. Sharma, Department of Mathematics, Central University of Jammu, Bagla,Rahya-Suchani, Samba 181143, INDIA
E-mail address : aksju [email protected] Sei-ichiro Ueki, Department of Mathematics, Faculty of Science, Tokai University,Hiratsuka 259-1292, JAPAN
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