The weighted composition operators on the large weighted Bergman spaces
aa r X i v : . [ m a t h . F A ] M a y THE WEIGHTED COMPOSITION OPERATORSON THE LARGE WEIGHTED BERGMAN SPACES
INYOUNG PARKA
BSTRACT . In this paper, we characterize bounded, compact or Schatten classweighted composition operators acting on Bergman spaces with the exponentialtype weights. Moreover, we give the proof of the necessary part for the bound-edness of C φ on large weighted Bergman spaces given by [8].
1. I
NTRODUCTION
Let φ be an analytic self-map of the open unit disk D in the complex plane and u be an analytic function on D . The weighted composition operator with respect to u is defined by ( uC φ ) f ( z ) := u ( z ) f ( φ ( z )) where f belongs to the holomorphicfunction space H ( D ) . There was much effort to characterize those analytic maps φ which induce bounded or compact weighted composition operators on the dif-ferent holomorphic function spaces (see for example, [3, 4, 6, 10, 13, 15]). When u ( z ) ≡ , it is well known that every composition operator is bounded on the stan-dard weighted Bergman spaces by the Littlewood subordination principle.On the other hand, in [8], Kriete and Maccluer showed that if there is a boundarypoint ζ such that the modulus of the angular derivative | φ ′ ( ζ ) | is less than thenthe map φ induces an unbounded operator C φ on the Bergman space with the fastweights. Moreover, they showed that we don’t know whether C φ is bounded or notwhen there is a boundary point | φ ′ ( ζ ) | = 1 by giving a explicit example map.In this paper, we study the boundedness, the compactness and schatten classweighted composition operators uC φ on the Bergman space with rapidly decreas-ing weights. For an integrable radial function ω , let L p ( ωdA ) be the space of allmeasurable functions f on D such that k f k pp,ω := Z D | f ( z ) | p ω ( z ) dA ( z ) < ∞ , < p < ∞ , where dA ( z ) is the normalized area measure on D . For a given < p < ∞ , theweighted Bergman space A p ( ω ) consists of f in the class of H ( D ) ∩ L p ( ωdA ) .Throughout this paper, we consider the radial weights of the form ω ( z ) = e − ϕ ( z ) such that ϕ ∈ C ( D ) and ∆ ϕ ( z ) ≥ C > for some constant C . When ϕ ( z ) = α log(1 − | z | ) where α > − , it represents the standard weighted Bergman spaces Mathematics Subject Classification.
Primary 30H20, 47B10, 47B35.
Key words and phrases. weighted Composition operator, Large weighted Bergman space, Schat-ten class. A pα ( D ) . In this paper, we assume that ϕ satisfies the following conditions (∆ ϕ ( z )) − =: τ ( z ) ց , and τ ′ ( z ) → when | z | → − so, we easily check that we exclude the standardweights since τ ( r ) = 1 − r when ϕ ( r ) = log(1 − r ) . In fact, these weightsdescribed above decrease faster than the standard weight (1 − | z | ) α , α > . Wecan refer to Lemma 2.3 of [12] for the proof. Furthermore, we assume that τ ( z )(1 −| z | ) − C increases for some C > or lim | z |→ − τ ′ ( z ) log τ ( z ) = 0 . Now, we saythat the weight ω is in the class W if ϕ satisfies all conditions above. If ω belongsto the class W then the associated function τ ( z ) has the following properties ( L ) :(i) There exists a constant C > such that τ ( z ) ≤ C (1 − | z | ) , for z ∈ D . (ii) There exists a constant C > such that | τ ( z ) − τ ( ξ ) | ≤ C | z − ξ | , for z, ξ ∈ D . The typical weight in the class W is the type of ω ( r ) = (1 − r ) β exp (cid:18) − c (1 − r ) α (cid:19) , β ≥ , α, c > , and its associated function τ ( z ) = (1 − | z | ) α . We can find more examples ofour weight functions and their τ functions in [12]. Additionally, we assume thatfast weights ω ∈ W have the regularity imposing the rate of decreasing conditionin [8], lim t → ω (1 − δt ) ω (1 − t ) = 0 , < δ < . (1.1)In [8], Kriete and Maccluer gave the following sufficient condition for the bound-edness of C φ and the equivalent conditions for the compactness of C φ on A ( ω ) . Theorem 1.1.
Let ω be a regular weight in the class W . If lim sup | z |→ ω ( z ) ω ( φ ( z )) < ∞ , (1.2) then C φ is bounded on A ( ω ) . Moreover, the following conditions are equivalent: (1) C φ is compact on A ( ω ) . (2) lim | z |→ ω ( z ) ω ( φ ( z )) = 0 . (3) d φ ( ζ ) > , ∀ ζ ∈ ∂ D where d φ ( ζ ) = lim inf z → ζ −| φ ( z ) | −| z | . In the same paper, they show that (1.2) is even the equivalent condition for theboundedness of C φ on A ( ω ) when ω ( z ) = e − −| z | ) α , α > . Therefore, we canexpect that the condition (1.2) is still the equivalent condition for the boundednessof C φ on the Bergman spaces with weights ω in the class W . In fact, there is almostthe converse of Theorem 1.1 in [5, Corollary 5.10] as follows: HE WEIGHTED COMPOSITION OPERATORS 3
Theorem 1.2. If lim inf r → ω ( r ) ω ( M φ ( r )) = ∞ then C φ is unbounded on A ( ω ) . Here, M φ ( r ) = sup θ | φ ( re iθ ) | . In Section 3, we show that the condition (1.2) is a necessary condition of theboundedness of C φ on A ( ω ) , thus we have the following result for general p withthe help of the Carleson measure theorem. Theorem 1.3.
Let ω be a regular weight in the class W and φ be an analytic self-map of D . Then the composition operator C φ is bounded on A p ( ω ) , < p < ∞ ifand only if lim sup | z |→ − ω ( z ) ω ( φ ( z )) < ∞ . In Section 4, we studied the weighted composition operator uC φ on A ( ω ) . In[6], we can find the characterizations of the boundedness, compactness and mem-bership in the Schatten classes of uC φ defined on the standard weighted Bergmanspaces in terms of the generalized Berezin transform. Using the same technique,we extended their results in [6] to the case of large weighted Bergman spaces A ( ω ) . Those our results are given by Theorem 4.3 and Theorem 4.5 in Section 4.Finally, we studied the membership of uC φ in the Schatten classes S p , p > inSection 5. Constants.
In the rest of this paper, we use the notation X . Y or Y & X fornonnegative quantities X and Y to mean X ≤ CY for some inessential constant C > . Similarly, we use the notation X ≈ Y if both X . Y and Y . X hold.2. S OME PRELIMINARIES
In this section, we recall some well-known notions and collect related facts tobe used in our proofs in later section.2.1.
Carleson type measures.
Let τ be the positive function on D satisfying theproperties ( L ) introduced in Section 1. We define the Euclidean disk D ( δτ ( z )) having a center at z with radius δτ ( z ) . Let m τ := min(1 , C − , C − )4 , (2.1)where C , C are in the properties (i), (ii) of ( L ) . In [12], they show that for < δ < m τ and a ∈ D , τ ( a ) ≤ τ ( z ) ≤ τ ( a ) , for z ∈ D ( δτ ( a )) . (2.2)The following lemma is the generalized sub-mean value theorem for | f | p ω . I. PARK
Lemma 2.1. [12, Lemma 2.2]
Let ω = e − ϕ , where ϕ is a subharmonic function.Suppose the function τ satisfies properties ( L ) and τ ( z ) ∆ ϕ ( z ) ≤ C for someconstant C > . For β ∈ R and < p < ∞ , there exists a constant M ≥ suchthat | f ( z ) | p ω ( z ) β ≤ Mδ τ ( z ) Z D ( δτ ( z )) | f | p ω β dA for a sufficiently small δ > , and f ∈ H ( D ) . A positive Borel measure µ in D is called a (vanishing) Carleson measure for A p ( ω ) if the embedding A p ( ω ) ⊂ L p ( ωdµ ) is (compact) continuous where L p ( ωdµ ) := (cid:26) f ∈ M ( D ) (cid:12)(cid:12) Z D | f ( z ) | p ω ( z ) dµ ( z ) < ∞ (cid:27) , (2.3)and M ( D ) is the set of µ -measurable functions on D . Now, we introduce theCarleson measure theorem on A p ( ω ) given by [11]. Theorem 2.2 (Carleson measure theorem) . Let ω ∈ W and µ be a positive Borelmeasure on D . Then for < p < ∞ , we have (1) The embedding I : A p ( ω ) → L p ( ωdµ ) is bounded if and only if for asufficiently small δ ∈ (0 , m τ ) , we have sup z ∈ D µ ( D ( δτ ( z ))) τ ( z ) < ∞ . (2) The embedding I : A p ( ω ) → L p ( ωdµ ) is compact if and only if for asufficiently small δ ∈ (0 , m τ ) , we have lim | z |→ µ ( D ( δτ ( z ))) τ ( z ) = 0 . From the above theorem, we note a Carleson measure is independent of p so thatif µ is a Carleson measure on L ( ωdµ ) then µ is a Carleson measure on L p ( ωdµ ) for all p .2.2. The Reproducing Kernel.
Let the system of functions { e k ( z ) } ∞ k =0 be anorthonormal basis of A ( ω ) . It is well known that the reproducing kernel for theBergman space A ( ω ) is defined by K ( z, ξ ) = K z ( ξ ) = ∞ X k =0 e k ( z ) e k ( ξ ) . Unlike the standard weighted Bergman spaces, the explicit form of K ( z, ξ ) of A ( ω ) has been unknown. However, we have the precise estimate near the diagonalgiven by Lemma 3.6 in [9], | K ( z, ξ ) | ≈ K ( z, z ) K ( ξ, ξ ) ≈ ω ( z ) − ω ( ξ ) − τ ( z ) τ ( ξ ) , ξ ∈ D ( δτ ( z )) , (2.4) HE WEIGHTED COMPOSITION OPERATORS 5 where δ ∈ (0 , m τ / . Recently, in [1], they introduced the upper estimate for thereproducing kernel as follows: for z, ξ ∈ D there exist constants C, σ > suchthat | K ( z, ξ ) | ω ( z ) / ω ( ξ ) / ≤ C τ ( z ) τ ( ξ ) exp (cid:18) − σ inf γ Z | γ ′ ( t ) | τ ( γ ( t )) dt (cid:19) , (2.5)where γ is a piecewise C curves γ : [0 , → D with γ (0) = z and γ (1) = ξ . Inthe same paper, they remarked that the distance β ϕ denoted by β ϕ ( z, ξ ) = inf γ Z | γ ′ ( t ) | τ ( γ ( t )) dt is a complete distance because of the property (i) of ( L ) . By the completeness of β ϕ and the kernel estimate (2.5), we obtain the following lemma. Lemma 2.3.
Let ω ∈ W . Then the normalized kernel function k z ( ξ ) uniformlyconverges to on every compact subsets of D when | z | → − .Proof. Given a compact subset K of D and z ∈ D , (2.4) and (2.5) follow that | k z ( ξ ) | = | K z ( ξ ) |k K z k ,ω . ω ( ξ ) − / τ ( ξ ) e − σβ ϕ ( z,ξ ) ≤ C K e − σβ ϕ ( z,ξ ) , ∀ ξ ∈ K . By the completeness of β ϕ , we conclude that k z ( ξ ) converges to uniformly oncompact subsets of D when z approaches to the boundary. (cid:3) The Julia-Caratheodory Theorem.
For a boundary point ζ and α > , wedefine the nontangential approach region at ζ by Γ( ζ, α ) = { z ∈ D : | z − ζ | < α (1 − | z | ) } . A function f is said to have a nontangential limit at ζ if ∠ lim z → ζz ∈ Γ( ζ,α ) f ( z ) < ∞ , for each α > . Definition 2.4.
We say φ has a finite angular derivative at a boundary point ζ ifthere is a point η on the circle such that φ ′ ( ζ ) := ∠ lim z → ζz ∈ Γ( ζ,α ) φ ( z ) − ηz − ζ < ∞ , for each α > . Theorem 2.5 (Julia-Caratheodory Theorem) . For φ : D → D analytic and ζ ∈ ∂ D ,the following is equivalent: (1) d φ ( ζ ) = lim inf z → ζ −| φ ( z ) | −| z | < ∞ . (2) φ has a finite angular derivative φ ′ ( ζ ) at ζ . (3) Both φ and φ ′ have (finite) nontangential limits at ζ , with | η | = 1 for η = lim r → φ ( rζ ) .Moreover, when these conditions hold, we have φ ′ ( ζ ) = d φ ( ζ )¯ ζη and d φ ( ζ ) = ∠ lim z → ζ −| φ ( z ) | −| z | . I. PARK
In addition to the Julia-Caratheodory theorem, we use the Julia’s lemma whichgives a useful geometric result. For k > and ζ ∈ ∂ D , let E ( ζ, k ) = { z ∈ D : | ζ − z | ≤ k (1 − | z | ) } . A computation shows that E ( ζ, k ) = n z ∈ D : (cid:12)(cid:12)(cid:12) k +1 ζ − z (cid:12)(cid:12)(cid:12) ≤ k k o . Theorem 2.6 (Julia’s Lemma) . Let ζ be a boundary point and φ : D → D beanalytic. If d φ ( ζ ) < ∞ then | φ ( ζ ) | = 1 where lim n →∞ φ ( a n ) =: φ ( ζ ) and { a n } is a sequence along which this lower limit is achieved. Moreover, φ ( E ( ζ, k )) ⊆ E ( φ ( ζ ) , kd φ ( ζ )) for every k > , that is, | φ ( ζ ) − φ ( z ) | − | φ ( z ) | ≤ d φ ( ζ ) | ζ − z | − | z | for all z ∈ D . (2.6)Note that (2.6) shows that d φ ( ζ ) = 0 if and only if φ is a unimodular con-stant. Moreover, when d φ ( ζ ) ≤ , φ ( E ( ζ, k )) ⊆ E ( φ ( ζ ) , k ) thus, the set E ( ζ, k ) contains its image of φ when φ ( ζ ) = ζ .2.4. Schatten Class.
For a positive compact operator T on a separable Hilbertspace H , there exist orthonormal sets { e k } in H such that T x = X k λ k h x, e k i e k , x ∈ H, where the points { λ k } are nonnegative eigenvalues of T . This is referred to as thecanonical form of a positive compact operator T . For < p < ∞ , a compactoperator T belongs to the Schatten class S p on H if the sequence { λ k } belongs tothe sequence space l p , k T k pS p = X k | λ k | p < ∞ . When ≤ p < ∞ , S p is the Banach space with the above norm and S p is a metricspace when < p < . In general, if T is a compact linear operator on H , we saythat T ∈ S p if ( T ∗ T ) p/ ∈ S , < p < ∞ . Moreover, ( T ∗ T ) p/ ∈ S ⇐⇒ T ∗ T ∈ S p/ . (2.7)In particular, when T ∈ S , we say that T is a Hilbert-Schmidt integral operator.It is well known that every Hilbert-Schmidt operator on L ( X, µ ) is an integraloperator induced by a function K ∈ L ( X × X, µ × µ ) such that T f ( z ) = Z X K ( x, y ) f ( y ) dµ ( y ) , ∀ f ∈ L ( X, dµ ) . The converse is also true. To study all of the basic properties of Schatten classoperators above, we can refer to § . and § . in [16]. HE WEIGHTED COMPOSITION OPERATORS 7
3. C
OMPOSITION O PERATORS C φ ON A p ( ω ) In [8], they also gave the necessary condition for the boundedness of C φ on largeBergman spaces in terms of the angular derivative, d φ ( ζ ) = lim inf z → ζ − | φ ( z ) | − | z | ≥ , ∀ ζ ∈ ∂ D . (3.1)The following Lemma enables us to consider only the boundary points d φ ( ζ ) = 1 in the proof of Theorem 3.2. Lemma 3.1.
Let ω be a regular fast weight. Then if d φ ( ζ ) > for ζ ∈ ∂ D then lim z → ζ ω ( z ) ω ( φ ( z )) = 0 . Proof.
We first assume that lim inf z → ζ −| φ ( z ) | −| z | = d φ ( ζ ) = 1 + 2 ǫ < ∞ for some ζ ∈ ∂ D . Then for ǫ > , there exists an open disk D r ( ζ ) centered at ζ with radius r such that − | φ ( z ) | − | z | > d φ ( ζ ) − ǫ, ∀ z ∈ D r ( ζ ) ∩ D . Thus, using the relation x A ≥ − A (1 − x ) when A > and < x < , we have | φ ( z ) | ≤ − ( d φ ( ζ ) − ǫ )(1 − | z | ) < − (1 + ǫ )(1 − | z | ) ≤ | z | ǫ , (3.2)for z ∈ D r ( ζ ) ∩ D . On the other hand, if we take a sufficiently small t such that (1 − t ) ǫ ≥ − ǫ ǫ t , then from the assumption (1.1) we obtain the followingresult, ω ( z ) ω ( | φ ( z ) | ) ≤ ω ( z ) ω ( | z | ǫ ) = ω ((1 − t ) ǫ ) ω (1 − t ) ≤ ω (1 − ǫ ǫ t ) ω (1 − t ) −→ , since ω ( | φ ( z ) | ) ≥ ω ( | z | ǫ ) by (3.2). When d φ ( ζ ) = lim inf z → ζ −| φ ( z ) | −| z | = ∞ ,we have the prompt result, | φ ( z ) | < | z | M , M > for near ζ by the definition,since there exists M > such that log 1 | φ ( z ) | > M log 1 | z | , for | z | near . Thus, we complete the proof. (cid:3) We now prove our first theorem.
Theorem 3.2.
Given a regular weight ω in W , let φ be an analytic self-map of D .If the composition operator C φ is bounded on A ( ω ) , then lim sup z → ζ ∈ ∂ D ω ( z ) ω ( φ ( z )) < ∞ . Proof.
Since the composition operator is bounded, we have d φ ( ζ ) ≥ for allboundary points ζ by (3.1). By Lemma 3.1, it suffices to check only the boundary I. PARK point ζ satisfying d φ ( ζ ) = 1 . For a boundary point ζ such that d φ ( ζ ) = 1 , we havethe following relation of inclusion by Theorem 2.6, φ (cid:18) E (cid:18) ζ, − r r (cid:19)(cid:19) ⊆ E (cid:18) φ ( ζ ) , − r r (cid:19) , r ∈ (0 , . Here, the set E ( ζ, − r r ) introduced in Section 2.3 is the closed disk centered at r ζ with radius − r so, rζ is the point on the boundary of E ( ζ, − r r ) closest to .Thus, we have (cid:12)(cid:12)(cid:12)(cid:12) r φ ( ζ ) − φ ( rζ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ − r , that is, | φ ( rζ ) | ≥ r for < r < since | φ ( ζ ) | = 1 by Theorem 2.5. Therefore, wehave r ∈ (0 , such that τ ( φ ( | z | ζ )) ≤ τ ( z ) for r ≤ | z | < , (3.3)since τ ( z ) is a radial decreasing function when | z | → − . Now, define the follow-ing radial function M φ ( r ) = sup η ∈ ∂ D | φ ( rη ) | , < r < . Then (3.3) follows that τ ( M φ ( | z | )) ≤ τ ( φ ( | z | ζ )) ≤ τ ( z ) , for r < | z | < . (3.4)Now, we assume that there is a sequence { z n } converges to ζ with ω ( z n ) ω ( φ ( z n )) → ∞ when n → ∞ . We can choose { ξ n } in D such that | ξ n | = | z n | and | φ ( ξ n ) | = M φ ( | z n | ) . Using the relation C ∗ φ K z ( ξ ) = h C ∗ φ K z , K ξ i = h K z , C φ K ξ i = K φ ( z ) ( ξ ) ,we have k C ∗ φ k z k ,ω = K φ ( z ) ( φ ( z )) k K z k ,ω ≈ τ ( z ) τ ( φ ( z )) ω ( z ) ω ( φ ( z )) , ∀ z ∈ D , thus by (3.4), we obtain k C φ k & k C ∗ φ k ξ n k ,ω & ω ( ξ n ) ω ( φ ( ξ n )) τ ( ξ n ) τ ( φ ( ξ n )) = ω ( z n ) ω ( M φ ( | z n | )) τ ( z n ) τ ( M φ ( | z n | )) ≥ ω ( z n ) ω ( φ ( z n )) , for large n > N . Thus, we have its contradiction and complete our proof. (cid:3) From Theorem 3.2 together with Theorem 1.1[KM], we conclude that (1.2) isthe equivalent condition for the boundedness of C φ on A ( ω ) . Furthermore, by thechange of variables in the measure theory in Theorem C in § of [7], we have k C φ f k pp,ω = Z D | f ◦ φ ( z ) | p ω ( z ) dA ( z ) = Z D | f ( z ) | p [ ωdA ] ◦ φ − ( z ) . Denote dµ φω ( z ) := ω ( z ) − [ ωdA ] ◦ φ − ( z ) . HE WEIGHTED COMPOSITION OPERATORS 9
Then C φ is (compact) bounded on A p ( ω ) for < p < ∞ if and only if the measure µ φω is a (vanishing) Carleson measure. Therefore the Carleson measure theoremshows that the condition (1.2) is still valid for the boundedness for all ranges of p . Theorem 3.3.
Given a regular weight ω in W , let φ be an analytic self-map of D .Then the composition operator C φ is bounded on A p ( ω ) , < p < ∞ if and only if lim sup | z |→ ω ( z ) ω ( φ ( z )) < ∞ . (3.5)4. W EIGHTED C OMPOSITION O PERATORS uC φ ON A ( ω ) For a positive Borel measure µ , we introduce the Berezin transform e µ on D given by e µ ( z ) = Z D | k z ( ξ ) | ω ( ξ ) dµ ( ξ ) , z ∈ D , where k z is the normalized kernel of A ( ω ) . For δ ∈ (0 , , the averaging function b µ δ over the disks D ( δτ ( z )) is defined by b µ δ ( z ) = µ ( D ( δτ ( z ))) | D ( δτ ( z )) | , z ∈ D . In [2], we already calculated the following relation between the Berezin transformand the averaging function, b µ δ ( z ) ≤ C e µ ( z ) , z ∈ D . (4.1)Moreover, we can see that (2),(3), and (4) in the following Proposition are equiv-alent in [2]. Before proving Proposition 4.2, we introduce the covering lemmawhich plays an essential role in the proof of many theorems including Carlesonmeasure theorem. Lemma 4.1. [11]
Let τ be a positive function satisfying properties ( L ) and δ ∈ (0 , m τ ) . Then there exists a sequence of points { a k } ⊂ D such that (1) a j / ∈ D ( δτ ( a k )) , for j = k . (2) S k D ( δτ ( a k )) = D . (3) e D ( δτ ( a k )) ⊂ D (3 δτ ( a k )) , where e D ( δτ ( a k )) = S z ∈ D ( δτ ( a k )) D ( δτ ( z )) (4) { D (3 δτ ( a k )) } is a covering of D of finite multiplicity N .Here, we call the sequence { a k } ∞ k =1 δ -sequence. Now, we characterize Carleson measures on A ( ω ) in terms of averaging func-tion and Berezin transform. Proposition 4.2.
Let ω ∈ W and δ ∈ (0 , m τ / . For µ ≥ , the followingconditions are equivalent: (1) µ is a Carleson measure on A ( ω ) . (2) e µ is bounded on D . (3) b µ δ is bounded on D . (4) The sequence { b µ δ ( a k ) } is bounded for every δ - sequence { a k } . Proof.
It is clear that (1) ⇒ (2). Thus, we remain to show that (4) implies (1). ByLemma 2.1, (3) of Lemma 4.1 and (2.2), we have sup ξ ∈ D ( δτ ( a k )) | f ( ξ ) | ω ( ξ ) . sup ξ ∈ D ( δτ ( a k )) τ ( ξ ) Z D ( δτ ( ξ )) | f | ωdA . τ ( a k ) Z D (3 δτ ( a k )) | f | ωdA. Therefore, by Lemma 4.1 and the inequality above, we have the desired result, Z D | f ( ξ ) | ω ( ξ ) dµ ( ξ ) ≤ ∞ X k =1 Z D ( δτ ( a k )) | f ( ξ ) | ω ( ξ ) dµ ( ξ ) ≤ ∞ X k =1 µ ( D ( δτ ( a k ))) sup ξ ∈ D ( δτ ( a k )) | f ( ξ ) | ω ( ξ ) . ∞ X k =1 µ ( D ( δτ ( a k ))) | D ( δτ ( a k )) | Z D (3 δτ ( a k )) | f | ωdA . sup a ∈ D b µ δ ( a ) N k f k ,ω . (4.2) (cid:3) For an analytic self-map φ of D and a function u ∈ L ( D ) , we define the φ -Berezin transform of u by B φ u ( z ) = Z D | k z ( φ ( ξ )) | u ( ξ ) ω ( ξ ) dA ( ξ ) . Theorem 4.3.
Let ω ∈ W and u be an analytic function on D . Then the weightedcomposition operator uC φ is bounded on A ( ω ) if and only if B φ ( | u | ) ∈ L ∞ ( D ) .Moreover, k uC φ k ≈ sup z ∈ D B φ ( | u | )( z ) .Proof. Given an analytic function u , we let dµ | u | ( z ) = | u ( z ) | ω ( z ) dA ( z ) . Now,we define the positive measure dµ φω,u ( z ) := ω ( z ) − dµ | u | ◦ φ − ( z ) . (4.3)By the change of variables in the measure theory, we have Z D | u ( z )( f ◦ φ )( z ) | ω ( z ) dA ( z ) = Z D | ( f ◦ φ )( z ) | dµ | u | ( z )= Z D | f ( z ) | ω ( z ) dµ φω,u ( z ) Thus, the fact that the measure dµ φω,u is a Carleson measure is equivalent to thatthe Berezin transform g µ φω,u is bounded on D by Proposition 4.2. On the other hand, g µ φω,u ( z ) = Z D | k z ( ξ ) | ω ( ξ ) dµ φω,u ( ξ )= Z D | k z ( φ ( ξ )) | | u ( ξ ) | ω ( ξ ) dA ( ξ ) = B φ ( | u | )( z ) , (4.4) HE WEIGHTED COMPOSITION OPERATORS 11 for all z ∈ D . Thus, the proof is complete. Furthermore, from (4.2) and (4.1), wehave k uC φ k . sup z ∈ D B φ ( | u | )( z ) . Finally, we obtain k uC φ k ≈ sup z ∈ D B φ ( | u | )( z ) since B φ ( | u | )( z ) ≤ k uC φ k forall z ∈ D . (cid:3) As an immediate consequence of Theorem 4.3 we obtain more useful necessarycondition for the boundedness of uC φ . Corollary 4.4.
Let ω ∈ W and u be an analytic function D . If uC φ is bounded on A ( ω ) , then sup z ∈ D τ ( z ) τ ( φ ( z )) ω ( z ) / ω ( φ ( z )) / | u ( z ) | < ∞ . (4.5) Proof.
For any z ∈ D , Theorem 4.3 follows that ∞ > B φ ( | u | )( φ ( z )) = Z D | k φ ( z ) ( φ ( ξ )) | | u ( ξ ) | ω ( ξ ) dA ( ξ ) ≥ Z D ( δτ ( z )) | k φ ( z ) ( φ ( ξ )) | | u ( ξ ) | ω ( ξ ) dA ( ξ ) & τ ( z ) | k φ ( z ) ( φ ( z )) | | u ( z ) | ω ( z ) & τ ( z ) τ ( φ ( z )) ω ( z ) ω ( φ ( z )) | u ( z ) | . (cid:3) Before characterizing the compactness of uC φ , we recall the definition of theessential norm k T k e for a bounded operator T on a Banach space X as follows: k T k e = inf {k T − K k : K is any compact operator on X } . For a bounded operator uC φ on A ( ω ) , we use the following formula introducedin [14] to estimate the essential norm of uC φ , k uC ϕ k e = lim n →∞ k uC ϕ R n k , (4.6)where R n is the orthogonal projection of A ( ω ) onto z n A ( ω ) defined by R n f ( z ) = ∞ X k = n a k z k for f ( z ) = ∞ X k =0 a k z k . The formula (4.6) can be obtained by the similar proof of Proposition 5.1 in [14].In fact, we can find the proof of (4.6) adjusting to large weighted Bergman spacesin [8, p.775]. Now, we prove the following estimate for the essential norm of abounded weighted composition operator uC φ on A ( ω ) . Theorem 4.5.
Let ω ∈ W and u be an analytic function on D . If uC φ is boundedon A ( ω ) , then there is an absolute constant C ≥ such that lim sup | z |→ B φ ( | u | )( z ) ≤ k uC φ k e ≤ C lim sup | z |→ B φ ( | u | )( z ) . Proof.
For k f k ,ω ≤ and a fixed < r < , we have k ( uC φ R n ) f k ,ω = Z D | R n f ( ξ ) | ω ( ξ ) dµ φω,u ( ξ )= Z D \ r D | R n f ( ξ ) | ω ( ξ ) dµ φω,u ( ξ ) + Z r D | R n f ( ξ ) | ω ( ξ ) dµ φω,u ( ξ ) , where r D = { z ∈ D : | z | ≤ r } . By the orthogonality of R n , we have | R n f ( ξ ) | ≤k f k ,ω k R n K ξ k ,ω and the following series is uniformly bounded on | ξ | ≤ r < , k R n K ξ k ,ω . ∞ X k = n p k r k < ∞ , where p k = Z s k +1 ω ( s ) ds. Thus, k R n K ξ k → as n → ∞ so that | R n f ( ξ ) | also uniformly converges to on r D as n → ∞ . Therefore the second integral vanishes as n → ∞ since µ φω,u is aCarleson measure. For the first integral, we denote by µ φω,u,r := µ φω,u | D \ r D . For afixed r > , we easily calculate ( D \ r D ) ∩ D ( δτ ( z )) = ∅ for | z | + δτ ( z ) > r .Thus by (4.1) and (4.2), the first integral is dominated by Z D | R n f | ω dµ φω,u,r . sup z ∈ D \ µ φω,u,rδ ( z ) k R n f k ,ω ≤ sup | z | + δτ ( z ) >r d µ φω,uδ ( z ) . sup | z | + δτ ( z ) >r g µ φω,u ( z ) , for all n > . Here, letting r → , then | z | → − since τ ( z ) → . Thus we obtainthe following upper estimate by (4.6) and the above inequality, k uC φ k e = lim n →∞ sup k f k ,ω ≤ k ( uC φ R n ) f k ,ω . lim sup | z |→ g µ φω,u ( z )= lim sup | z |→ B φ ( | u | )( z ) . For the lower estimate, for any compact operator K on A ( ω ) , we have k uC φ k e ≥ k uC φ − K k ≥ lim sup | z |→ k ( uC φ ) k z k ,ω = lim sup | z |→ B φ ( | u | )( z ) . (cid:3) In addition to Theorem 4.5, we have the following useful necessary conditionfor the compactness of uC φ on A ( ω ) . HE WEIGHTED COMPOSITION OPERATORS 13
Corollary 4.6.
Let ω ∈ W and u be an analytic function on D . If the weightedcomposition operator uC φ is compact on A ( ω ) then lim | z |→ − τ ( z ) τ ( φ ( z )) ω ( z ) ω ( φ ( z )) | u ( z ) | = 0 . (4.7) Proof.
By Lemma 2.3, we know that the normalized kernel sequence { k z } con-verges to weakly when | z | → . Since ( uC φ ) ∗ K z = u ( z ) K φ ( z ) , we have | z |→ k ( uC φ ) ∗ k z k ,ω = lim | z |→ | u ( z ) | K ( φ ( z ) , φ ( z )) K ( z, z ) & lim | z |→ | u ( z ) | τ ( z ) τ ( φ ( z )) ω ( z ) ω ( φ ( z )) . (cid:3)
5. S
CHATTEN CLASS WEIGHTED COMPOSITION OPERATORS
In order to study the Schatten class composition operator, we use some knownresults of Toeplitz operators acting on A ( ω ) . First of all, we recall that the defini-tion and some facts of Toeplitz operators defined on A ( ω ) . For a finite complexBorel measure µ on D , the Toeplitz operator T µ is defined by T µ f ( z ) = Z D f ( ξ ) K ( z, ξ ) ω ( ξ ) dµ ( ξ ) , for f ∈ A ( ω ) . Here, it is not clear the integrals above will converge, so we givethe following additional condition on µ , Z D | K ( z, ξ ) | ω ( ξ ) d | µ | ( ξ ) < ∞ , (5.1)for all z ∈ D . The following lemma is from Lemma 2.2 in [2]. Lemma 5.1.
Let ω ∈ W , µ ≥ and µ be a Carleson measure. Then we have h T µ f, g i ω = Z D f ( ξ ) g ( ξ ) ω ( ξ ) dµ ( ξ ) , f, g ∈ A ( ω ) , where h f, g i ω = R D f ( z ) g ( z ) ω ( z ) dA ( z ) . It is well known that composition operators are closely related to Toeplitz oper-ators on weighted Bergman spaces. From Lemma 5.1, we have the relation, h ( uC φ ) ∗ ( uC ϕ ) f, g i ω = Z D f ( φ ( z )) g ( φ ( z )) | u ( z ) | ω ( z ) dA ( z )= Z D f ( z ) g ( z ) ω ( z ) dµ φω,u ( z ) = h T µ φω,u f, g i ω , (5.2)where dµ φω,u is defined by (4.3). Thus, we can use the results of T µ φω,u to see whenthe composition operators uC ϕ belong to S p . In other words, it suffices to showwhen T µ φω,u is in S p/ since uC ϕ ∈ S p is equivalent to ( uC ϕ ) ∗ ( uC ϕ ) ∈ S p/ as westudied in Section 2.4. The following lemma is the characterization of the membership in the Schattenideals of a Toeplitz operator acting on A ( ω ) . Lemma 5.2 (Theorem 4.6 [2]) . Let ω ∈ W , δ ∈ (0 , m τ / and < p < ∞ . For µ ≥ , the following conditions are equivalent: (1) T µ ∈ S p ( A ( ω )) . (2) e µ ∈ L p (∆ ϕdA ) . (3) b µ δ ∈ L p (∆ ϕdA ) . Theorem 5.3.
Let < p < ∞ . Let ω ∈ W and u be an analytic function on D .Then uC φ ∈ S p if and only if B φ ( | u | ) ∈ L p/ (∆ ϕdA ) .Proof. If uC φ ∈ S p then ( uC ϕ ) ∗ ( uC ϕ ) = T µ φω,u ∈ S p/ by (5.2). Thus, we havethe equivalent condition g µ φω,u = B φ ( | u | ) ∈ L p/ (∆ ϕdA ) from (4.4) and Lemma5.2. (cid:3) Corollary 5.4.
Let u be an analytic function on D and let φ be an analytic self-mapof D . Then uC φ is a Hilbert-Schmidt operator if and only if Z D τ ( z ) τ ( φ ( z )) ω ( z ) ω ( φ ( z )) | u ( z ) | ∆ ϕ ( z ) dA ( z ) < ∞ . Proof.
By Theorem 5.3 and (2.4), we have Z D | B φ ( | u | )( z ) | ∆ ϕ ( z ) dA ( z )= Z D Z D | k z ( φ ( ξ )) | | u ( ξ ) | ω ( ξ ) dA ( ξ ) 1 τ ( z ) dA ( z ) ≈ Z D Z D | K ( z, φ ( ξ )) | | u ( ξ ) | ω ( ξ ) dA ( ξ ) ω ( z ) dA ( z )= Z D K ( φ ( ξ ) , φ ( ξ )) | u ( ξ ) | ω ( ξ ) dA ( ξ ) ≈ Z D τ ( ξ ) τ ( φ ( ξ )) ω ( ξ ) ω ( φ ( ξ )) | u ( ξ ) | ∆ ϕ ( ξ ) dA ( ξ ) < ∞ . (cid:3) Acknowledgements.
The author would like to thank the referee for indicatingvarious mistakes and giving helpful comments.R
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