Weighted composition operators on weighted Bergman spaces of bounded symmetric domains

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1 Proc. Indian Acad. Sci. (Math. Sci.) Vol. 117, No. 2, May 2007, pp Printed in India Weighted composition operators on weighted Bergman spaces of bounded symmetric domains SANJAY KUMAR and KANWAR JATINDER SINGH Department of Mathematics, University of Jammu, Jammu , India sks MS received 7 December 2005 Abstract. In this paper, we study the weighted compositon operators on weighted Bergman spaces of bounded symmetric domains. The necessary and sufficient conditions for a weighted composition operator W ϕ,ψ to be bounded and compact are studied by using the Carleson measure techniques. In the last section, we study the Schatten p-class weighted composition operators. Keywords. Bergman spaces; bounded symmetric domain; boundedness; Carleson measure; compactness; Schatten p-class operator. 1. Introduction Let be a bounded symmetric domain in C n with Bergman kernel K(,ω). We assume that is in its standard representation and the volume measure dv of is normalised so that K(,0) K(0,ω),for all and ω in. By Theorem 5.7 of [11] and using the polar coordinates representation, there exists a positive number ε such that for <ε, we have C K(, ) dv()<. For each <ε, define dv () C 1 K(, ) dv().then {dv } defines a weighted family of probability measures on. Also, throughout the paper <ε is fixed. We define the weighted Bergman spaces A p (, dv ), on, as the set of all holomorphic functions f on so that ) 1 f p A f ( p p dv <. Note that A p (, dv ) is a closed subspace of L p (, dv ). For 0,A p (, dv ) is just the usual Bergman space. For p 2, there is an orthogonal projection P from L 2 (, dv ) onto A 2 (, dv ) given by P f() K (, ω)f (ω)dv (ω), where K (, ω) K(,ω) 1 is the reproducing kernel for A 2 (, dv ). 185

2 186 Sanjay Kumar and Kanwar Jatinder Singh Suppose, ϕ,ψ are holomorphic mappings defined on such that ϕ(). Then the weighted composition operator W ϕ,ψ is defined as W ϕ,ψ f() ψ()f (ϕ()), for all f holomorphic in, and. For the study of weighted composition operators one can refer to [4, 6, 13, 15] and references therein. Recently, Smith [16] has made a nice connection between the Brennan s conjecture and weighted composition operators. He has shown that Brennan s conjecture is equivalent to the existence of self-maps of unit disk that make certain weighted composition operators compact. We know that the bounded symmetric domain in its standard representation with normalised volume measure, the kernel function K(, ) has the following special properties: (1) K(0,a) 1 K(a,0). (2) K(,a) 0 (for all in and a in ). (3) lim a K(a,a). (4) K(,a) 1 is a smooth function on C n C n. Here denotes the closure of in C n and is the topological boundary, see [8] and [10] for details. For any a in, let k a () K(,a) K(a,a). The k a s are called normalised reproducing kernels for A2 (, dv).they are unit vectors in A 2 (, dv).moreover, ka 1 is a unit vector of A 2 (, dv ) for any a in. Let µ be a finite complex Borel measure on. Then the Berein transform of measure µ, denoted by µ, is defined as µ () k (ω) 2(1 ) dµ(ω),. We will denote by β(,ω) the Bergman distance function on. For and r>0, let {ω : β(,ω)<r}. We denote by the normalised volume of, that is, dv(ω). Given a finite complex Borel measure µ on, we define a function ˆµ r on by ˆµ r () µ(e(, r)) 1,. Theorem 1.1 [17]. Take 1 p<. Let µ be a finite positive Borel measure on. Then the following conditions are equivalent: (1) The inclusion map i: A p (, dv ) L p (, dv ) is bounded. (2) The Berein transform µ is bounded.

3 Bergman spaces of bounded symmetric domains 187 (3) ˆµ r is bounded on for all (or some) r>0. (4) {ˆµ r (a n )} is a bounded sequence, where {a n } is some bounded sequence in independent of µ. A positive Borel measure µ satisfying any one of the conditions of Theorem 1.1 is called a Carleson measure on the weighted Bergman space A p (, dv ) Theorem 1.2 [17]. Take 1 p<. Let µ be a finite positive Borel measure on. Then the following conditions are equivalent: (1) The inclusion map i: A p (, dv ) L p (, dv ) is compact. (2) The Berein transform µ () 0 as. (3) ˆµ r () 0 as for all (or some) r>0. (4) ˆµ r (a n ) 0 as n. A positive Borel measure µ satisfying any one of the conditions of Theorem 1.2 is called a vanishing Carleson measure on the weighted Bergman space A p (, dv ). A positive compact operator T on A 2 (, dv ) is in the trace class if tr(t ) Te n,e n <, n1 for some (or all) orthonormal basis {e n } of A 2 (, dv ). Take 1 p<, and T be a compact linear operator on A 2 (, dv ). Then we say that T belongs to the Schatten p-class S p if (T T) p/2 is in the trace class. Also, the S p norm of T is given as T Sp [tr((t T) p/2 )] 1/p. Moreover, S p is a two-sided ideal of the full algebra B(A 2 (, dv )) of bounded linear operators on A 2 (, dv ). 2. Preliminaries To make the paper self-contained we state the following lemmas. Lemma 2.1 [12]. The Bergman kernel K(,ω) is conjugate symmetric. That is, K(,ω) K(ω,); and it is the function of in A 2 (, dv ). Lemma 2.2 [17]. Let T be a trace class operator or a positive operator on A 2 (, dv ). Then tr(t ) TK 1,K 1 A 2 dσ(), where dσ() K(, )dv().

4 188 Sanjay Kumar and Kanwar Jatinder Singh Lemma 2.3 [9]. Let be a bounded symmetric domain in C n and let W ϕ,ψ be a weighted composition operator on A 2 (, dv ). Then W ϕ,ψ A 2 [ ] (1 ) K(ϕ(), ϕ()) ψ() 2. K(, ) Lemma 2.4 [1]. Suppose T is a positive operator on a Hilbert space H and x is a unit vector in H. Then (1) T p x,x Tx,x p for all p 1. (2) T p x,x Tx,x p for all 0 <p 1. Lemma 2.5 (Theorem 18.11(f), p. 89of [5]). Suppose T is a bounded linear operator on a Hilbert space H.IfT S p, then tr( T p ) tr( T p ). Lemma 2.6 [17]. Take 1 p<. Suppose µ 0 is a finite Borel measure on. Then µ is a Carleson measure on A p (, dv ) if and only if µ(e(, r))/ 1 is bounded on (as a function of ) for all (or some) r>0. Moreover, the following quantities are equivalent for any fixed r>0and p 1: { } µ(e(, r)) ˆµ r sup : 1 and { } µ p sup f() p dµ() f() p dv () : f Ap (, dv ). Lemma 2.7 [2]. For any r>0, there exists a constant C (depending only on r) such that C 1 E(a,r) k a () 2 C, for all a and E(a,r). Note that if we take a in the above estimate, then we get for all a. C 1 E(a,r) K(a,a) C, Lemma 2.8 [2]. For any r, s, R) such that r > 0, s > 0, R > 0, there exists a constant C (depending on C 1 E(a,r) E(a,s) C for all a,b in with β(a,b) R. Lemma 2.9 [3]. For any two conditions: r>0, there exists a sequence {a n } in satisfying the following

5 Bergman spaces of bounded symmetric domains 189 (1) n1 E(a n,r); (2) There is a positive integer N such that each point in belongs to at most N of the sets E(a n, 2r). Lemma 2.10 [3]. For any r>0and p 1, there exists a constant C (depending only on r) such that f(a) p C f() p dv(), E(a,r) E(a,r) for all f holomorphic and a in. 3. Bounded and compact weighted composition operators By using [7], we can prove the following lemma. Lemma 3.1. Take 1 p <. Let be a bounded symmetric domain in C n, and <ε. Suppose ϕ,ψ be holomorphic functions defined on such that ϕ(). Also, suppose that the weighted composition operator W ϕ,ψ is bounded on A p (, dv ). Then W ϕ,ψ is compact on A p (, dv ) if and only if for any norm bounded sequence {f n } in A p (, dv ) such that f n 0 uniformly on compact subsets of, then we have W ϕ,ψ f n A p 0. Lemma 3.2. Suppose {f n } is a sequence in A p (, dv ) such that f n 0 weakly in A p (, dv ). Then {f n } is a norm bounded sequence and f n 0 uniformly on each compact subsets of. Let ϕ be a holomorphic function defined on such that ϕ(). Suppose ψ A p (, dv ) and <ε. Then the nonnegative measure µ ϕ,ψ,p is defined as µ ϕ,ψ,p (E) ψ p dv, ϕ 1 (E) where E is a measurable subset of. Using Lemma 2.1 of [4], we can prove the following result. Lemma 3.3. Take 1 p<. Let be a bounded symmetric domain in C n, and <ε. Let ϕ be a holomorphic mapping from into and ψ A p (, dv ). Then gdµ ϕ,ψ,p ψ p (g ϕ)dv, where g is an arbitrary measurable positive function on. Theorem 3.4. Take 1 p<. Let be a bounded symmetric domain in C n, and <ε. Suppose ϕ,ψ are holomorphic functions defined on such that ϕ(). Then the weighted composition operator W ϕ,ψ is bounded on A p (, dv ) if and only if the measure µ ϕ,ψ,p is a Carleson measure.

6 190 Sanjay Kumar and Kanwar Jatinder Singh Proof. Suppose W ϕ,ψ is bounded on A p (, dv ). Then there exists a constant M>0 such that W ϕ,ψ f A p M f A p, for all f A p (, dv ). Take g(ω) k (ω) 2(1 ) p. From Lemma 2.1 and using the fact that K(ω,) 0 for any and ω in, we get that g A p (, dv ). Thus, we have k (ω) 2(1 ) dµ ϕ,ψ,p (ω) g(ω) p dµ ϕ,ψ,p (ω) ψ(ω) p g(ϕ(ω)) p dv (ω) W ϕ,ψ g p A p <. By Lemma 2.7, there exists a constant C>0 (depending only on r) such that µ ϕ,ψ,p (E(, r)) dµ ϕ,ψ,p (ω) C k (ω) 2(1 ) dµ ϕ,ψ,p (ω) <, for any. Thus, by using Lemma 2.6, we get that µ ϕ,ψ,p is a Carleson measure on A p (, dv ). Conversely, suppose µ ϕ,ψ,p is a Carleson measure on A p (, dv ). Then by Theorem 1.1, there exists a constant M>0such that f() p dµ ϕ,ψ,p () M p f p A p, for all f A p (, dv ). Hence, using Lemma 3.3, we have W ϕ,ψ f p A ψ() p f(ϕ()) p dv () f(ω) p dµ ϕ,ψ,p (ω) M p f(ω) p dv (ω) <, for all f A p (, dv ). Hence, W ϕ,ψ is a bounded operator on A p (, dv ).

7 Bergman spaces of bounded symmetric domains 191 Theorem 3.5. Take 1 p<. Let be a bounded symmetric domain in C n, and <ε. Suppose ϕ,ψ are holomorphic functions defined on such that ϕ(). Then the weighted composition operator W ϕ,ψ is compact on A p (, dv ) if and only if the measure µ ϕ,ψ,p is a vanishing Carleson measure. Proof. Suppose W ϕ,ψ is compact on A p (, dv ). Since, k 1 0 weakly in A p (, dv ) as, by Lemmas 3.1 and 3.2, we have W ϕ,ψ k 1 p A 0 as. (3.1) Again, by using Lemma 2.7, we get a constant C>0 (depending only on r) such that µ ϕ,ψ,p (E(, r)) dµ ϕ,ψ,p (ω) C k (ω) 2(1 ) dµ ϕ,ψ,p (ω) C C W ϕ,ψ k 1 A 2. ψ(ω) 2 k (ϕ(ω)) 2(1 ) dµ ϕ,ψ,p (ω) Thus, condition (3.1) implies that µ ϕ,ψ,p is a vanishing Carleson measure on A p (, dv ). Conversely, suppose that µ ϕ,ψ,p is a vanishing Carleson measure on A p (, dv ). Then, by Theorem 1.2, we have µ ϕ,ψ,p (E(, r)) lim 1 0 (3.2) for all (or some) r>0. So, from Theorem 3.4, we conclude that W ϕ,ψ is bounded on A p (, dv ). We will prove that W ϕ,ψ is a compact operator. Let {f n } be a sequence in A p (, dv ) and f n 0 weakly. By Lemma 3.2, f n is a norm bounded sequence and f n 0 uniformly on each compact subsets of. By Lemmas 2.7 and 2.10, there exists a constant C>0 (depending only on r) such that f() p C f(ω) p dv(ω) CC f(ω) p dv (ω) K(ω,ω) C C 1 1 f(ω) p dv (ω) E(ω,r) f(ω) p dv (ω),

8 192 Sanjay Kumar and Kanwar Jatinder Singh for all, where the last inequality is obtained by using Lemma 2.8. By combining the above inequality with Lemma 2.8, we get { } sup{ f() p C 1 : E(a,r)} sup 1 f(ω) p dv (ω): E(a,r) E(a,2r) C 2 E(a,r) 1 f(ω) p dv (ω), E(a,2r) where C 1 and C 2 are constants depending only on and r. Hence, we have W ϕ,ψ f n p A ψ() p f n (ϕ()) p dv (ω) f n (ω) p dµ ϕ,ψ,p C i1 E(a i,r) f n (ω) p dµ ϕ,ψ,p µ ϕ,ψ,p ((E(a i,r))sup{ f n (ω) p : ω E(a i,r)} i1 i1 µ ϕ,ψ,p ((E(a i,r)) E(a i,r) 1 E(a i,2r) f n (ω) p dv (ω), where the sequence {a n } is the same as in Lemma 2.9. From condition (3.2), for every ɛ>0, there exists a δ>0 (0 <δ<1) such that µ ϕ,ψ,p (E(, r)) 1 whenever d(, )<δ. Take <ɛ, 1 {ω : d(, ) δ, β(, ω) r}. Then, 1 is a compact subset of (as 0 < 2r < 2 δ ). Thus, we have W ϕ,ψ f n () p dv () f n (ω) p dµ ϕ,ψ,p (ω) C C + C i1 µ ϕ,ψ,p ((E(a i,r)) E(a i,r) 1 d(a, ) δ d(a, )<δ µ ϕ,ψ,p ((E(a i,r)) E(a i,r) 1 E(a i,2r) µ ϕ,ψ,p ((E(a i,r)) E(a i,r) 1 f n (ω) p dv (ω) E(a i,2r) E(a i,2r) f n (ω) p dv (ω) f n (ω) p dv (ω)

9 Bergman spaces of bounded symmetric domains 193 CN + Cɛ CNK d(a, ) δ d(a, )<δ E(a i,2r) E(a i,2r) f n (ω) p dv (ω) f n (ω) p dv (ω) f n (ω) p dv (ω) + CɛK f n (ω) p dv (ω), where the constant N is the same as in Lemma 2.9. Since, f n is a norm bounded sequence and f n converges to ero uniformly on compact subsets of, we have W ϕ,ψ f n p 0 A as n. The following theorem follows by combining Theorems 1.1, 1.2, 3.4 and 3.5. Theorem 3.6. Take 1 p< and let <ε. Suppose ϕ,ψ be holomorphic functions defined on such that ϕ(). Then (1) The weighted composition operator W ϕ,ψ is bounded on A p (, dv ) if and only if sup k (ω) 2(1 ) dµ ϕ,ψ,p (ω) <. (2) The weighted composition operator W ϕ,ψ is compact on A p (, dv ) if and only if lim k (ω) 2(1 ) dµ ϕ,ψ,p (ω) Schatten class weighted composition operators Theorem 4.1. Take 1 p< and let <ε. Suppose ϕ,ψ are holomorphic functions defined on such that ϕ(). Let W ϕ,ψ be compact on A 2 (, dv ). Then W ϕ,ψ S p if and only if the Berein symbol of measure µ ϕ,ψ,p belongs to L p/2 (, dσ), where dσ() K(, )dv(). Proof. For any f, g A 2 (, dv ), we have (W ϕ,ψ ) (W ϕ,ψ )f, g A 2 (W ϕ,ψ )f, (W ϕ,ψ )g A 2 f(ϕ())g(ϕ()) ψ() 2 dv () f(ω)g(ω) ψ() 2 dµ ϕ,ψ,p (ω). Consider the Toeplit operator T µϕ,ψ,p f() f(ω)k (, ω) dµ ϕ,ψ,p (ω).

10 194 Sanjay Kumar and Kanwar Jatinder Singh Then, by using Fubini s theorem, we have T µϕ,ψ,p f, g A 2 f(ω)k (, ω)dµ ϕ,ψ,p (ω)g() dv () f(ω) g()k (ω, )dv () dµ ϕ,ψ,p (ω) f(ω)g(ω) dµ ϕ,ψ,p (ω). Thus, T µϕ,ψ,p (W ϕ,ψ ) (W ϕ,ψ ). Also, by definition an operator T S p if and only if (T T) p/2 is in the trace class, and this is equivalent to saying that T T S p/2 (Lemma 1.4.6, p. 18 of [18]). Thus W ϕ,ψ S p if and only if T µϕ,ψ,p S p/2. Again, by Theorem C of [17], T µϕ,ψ,p S p/2 if and only if the Berein symbol of the measure µ ϕ,ψ,p belongs to L p/2 (, dσ). Theorem 4.2. Take 2 p< and let <ε. Take W ϕ,ψ the weighted composition operator on A 2 (, dv ). If W ϕ,ψ S p, then [ K(ϕ(), ϕ()) ψ() K(, ) where dσ() K(, )dv(). Proof. By using Lemmas , we get ] p(1 ) 2 dσ() <, tr( W ϕ,ψ p ) tr( Wϕ,ψ p ) tr( W ϕ,ψ Wϕ,ψ )p/2 (W ϕ,ψ Wϕ,ψ )p/2 k 1,k 1 A 2 dσ() (W ϕ,ψ Wϕ,ψ )k1 W ϕ,ψ k1 p A 2 dσ() [ K(ϕ(), ϕ()) ψ() K(, ),k 1 p/2 A 2 ] p(1 ) 2 dσ() dσ(). Theorem 4.3. Take 0 <p<2 and let <ε. Take W ϕ,ψ the weighted composition operator W ϕ,ψ on A 2 (, dv ). If [ K(ϕ(), ϕ()) ψ() K(, ) ] p(1 ) 2 where dσ() K(, )dv(),then W ϕ,ψ S p. dσ() <,

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