Compactness and Norm of the Sum of Weighted Composition Operators on A(D)

Transcription

1 Int. Journal of Math. Analysis, Vol. 4, 2010, no. 39, Compactness and Norm of the Sum of Weighted Composition Operators on A(D) Harish Chandra and Bina Singh Department of Mathematics and DST-CIMS Banaras Hindu University, Varanasi, India harish Abstract. Let A(D) denote the disk algebra and uc ϕ be the weighted composition operator on A(D). In this paper, we characterize the compactness of sum of weighted composition operators on A(D). Furthermore, we also find the norm of sum of weighted composition operators on A(D). Mathematics Subject Classification: 47B33 Keywords: composition operator, weighted composition operator, pseudohyberbolic metric, compact operator 1. Introduction Let D denote the standard unit disk in the complex plane C and D be its closure in C. Let H = H (D) be the space of all bounded analytic functions on the open unit disk D. Then H is a Banach algebra with the supremum norm f = sup { f (z) : z D}. The disk algebra A(D) is the Banach algebra of all continuous functions on D which are analytic in D, under the supremum norm f = sup { f (z) : z D }. By the maximum modulus principle, we can see that this norm is equal to f. For functions u and ϕ A(D) with ϕ 1, we define a weighted composition operator uc ϕ on A(D) by (uc ϕ f)(z) =u(z)f(ϕ(z)) z D. It is easy to see that uc ϕ is a bounded linear operator and uc ϕ = u. When u (z) 1, we just have the composition operator C ϕ on A(D). For z and w D, the pseudo-hyperbolic distance between z and w is given by

2 1946 H. Chandra and B. Singh z w ρ (z, w) = 1 zw, if z w 0, if z = w. Weighted composition operators appear in the literature in different ways. For example, the endomorphisms of semi-simple commutative Banach algebras can be represented as weighted composition operators. Hence these operators, and also their sums, are interesting to study from different point of views such as compactness, norm and so on. The disk algebra is an illuminating example of semi-simple commutative Banach algebra. Therefore it is instructive to study weighted composition operators and their sums on it. The aim of this paper is to clarify the compactness and norm of sum of weighted composition operators on the disk algebra. The various properties of weighted composition operators have been extensively studied on Hardy spaces H p into itself in [2], [12]. In the H setting, some results were obtained in [3], [13]. These operators have been studied in other spaces of analytic functions as well (See for [11], [1], [8], [9] and references therein). Operator theoretic properties of a weighted composition operator depends canonically on function theoretic properties of u and ϕ. Consequently, the study of weighted composition operator uc ϕ on the spaces of analytic functions is a meeting point between functional analysis and analytic function theory. This interplay has resulted in deep and profound theorems which have illuminated the whole of operator theory. This apart, these operators are also source of rich variety of examples of linear operators. In 1979, Herbert Kamowitz characterized the compactness of weighted composition operators on A(D) and studied their spectra [10]. We state his result as follows. Theorem 1.1. Let u, ϕ in A(D) with ϕ 1 and suppose ϕ is a non-constant function. Then uc ϕ is a compact operator on A(D) if and only if ϕ(z) < 1 whenever u(z) 0. In [6] Gorkin and Mortini studied norms and essential norms of finite linear combinations of composition operators on uniform algebras. Further, Izuchi and S.Ohno have characterized compactness of finite linear combinations of compostion operators on H in [9]. In this article we study properties of the sum of weighted composition operators on A(D). In section 2, we give a characterization of compact operators which are the sum of weighted composition operators on A(D). Furthermore, we also give a necessary and sufficient condition under which the norm of the infinite sum of weighted composition operators on A(D) is equal to the sum of norm of each component in section 3. In what follows, we denote by T the boundary of the unit disk D and S(D) the set of all analytic self map of D. We refer to the reader to [4], [5], [7] and [14] for the results about the Banach spaces of analytic functions. For function u H and ϕ S(D), we define a weighted composition operator uc ϕ on

3 Weighted composition operators 1947 H by uc ϕ f = u(f ϕ) for f H. It is clear that uc ϕ is linear and bounded on H and its properties were investigated in [3] and [13]. 2. compactness In this section we study the compactness of those operators which can be expressed as a sum of weighted composition operators on A(D). It is well known that a composition operator C ϕ on, say a Banach space X of analytic functions in the disk, is compact if and only if whenever {f m } is a bounded sequence in X and f m 0 uniformly on compact subsets of D, then C ϕ (f m ) X 0. It can be readly seen that the above result also holds for those compact operators which are sum of weighted composition operators on H. The particular instance of this result, which we will need, is the following proposition. Proposition 2.1. Let ϕ 1,ϕ 2,...,ϕ N be distinct functions in S(D) and u 1,u 2,...,u N in H with u i 0for every i. Then N u i C ϕi is compact on H if and only if whenever {f n } n is a bounded sequence in H such that {f n } n 0 uniformly N on every compact subsets of D then u i C ϕi f n 0 as n. Proof. The proof is the special case of the proof of proposition 3.11 in [4]. Let ϕ 1,ϕ 2,...,ϕ N be distinct functions in S(D) and N 2. Let Z = Z(ϕ 1,ϕ 2,...,ϕ N ) be the family of sequences {z n } n in D satisfying the following three conditions: (a) ϕ i (z n ) 1asn for some i, (b) {ϕ { i (z n )} n is convergent } sequence for every i, ϕ j (z n ) ϕ i (z n ) (c) is a convergent sequence for every i, j. 1 ϕ j (z n )ϕ i (z n ) n Condition (c) implies that {ρ(ϕ i (z n ),ϕ j (z n ))} n is a convergent sequence for every i, j. Note that if ϕ i (z n ) 1asn for some i, then it is easy to see that there exists a subsequence { } z of {z nj j n} n satisfying { z nj }j Z. For {z n } n Z, we write I({z n })={i :1 i N, ϕ i (z n ) 1asn }. By condition (a), I({z n }) ϕ. By(b), there exists δ with 0 <δ<1 such that ϕ j (z k ) <δ<1 for every j/ I({z n }) and for every k. For each t I ({z n }), we write (2.1) I 0 ({z n },t)={j I ({z n }):ρ(ϕ j (z n ),ϕ t (z n )) 0asn }. For s, t I ({z n }), we have either I 0 ({z n },s)=i 0 ({z n },t)ori 0 ({z n },s) I 0 ({z n },t)=ϕ. Hence there is a subset {t 1,t 2...,t l } I({z n }) such that I ({z n })= l p=1 I 0 ({z n },t p ) and I 0 ({z n },t p ) I 0 ({z n },t q )=ϕ, for p q.

4 1948 H. Chandra and B. Singh We begin with the following theorem which generalises the result given by Ohno and Izuchi in [9]. Theorem 2.1. Let ϕ 1,ϕ 2,...,ϕ N be distinct functions in S(D) and u 1,u 2,...,u N in H with u i 0for every i. Then the following conditions are equivalent. N (i) u i C ϕi is compact on H. (ii) u i (z n ) 0 as n for every {z n } n Z and t I ({z n }). i I 0 ({z n},t) Proof. The idea of the proof of this theorem is same as that of Theorem 2.2 in [9]. In the next theorem we show that compactness of finite sum of weighted compostition operator on A(D) is equivalent to that on H. Theorem 2.2. Let ϕ 1,ϕ 2,...,ϕ N be distinct functions in A(D) with ϕ i = 1 for each i and u 1,u 2,...,u N in A(D) with u i 0for every i. Then the following is equivalent. (i) N u i C ϕi is compact on H. (ii) N u i C ϕi is compact on A(D). Proof. (i) (ii). Suppose N u i C ϕi is compact on H. We claim that N u i C ϕi is compact on A(D). Let {f n } be a bounded sequence in A(D). Since A(D) H (, hence we get a subsequence {f nk } and a function g H such N ) that lim u i C ϕi f nk = g. Clearly g A(D). Thus N u i C ϕi is compact k on A(D). (ii) (i) Suppose that N u i C ϕi is not compact on H. Then for some ɛ>0 there exists ( {f n } n ball H such that f n 0 uniformly on compact subset N ) of D and u i C ϕi f n >ɛfor all n. Hence there is {z n } n D such N that u i (z n )f n (ϕ i (z n )) >ɛ.for 0 <r<1, define f n,r(z) =f n (rz). Then f n,r ball A(D). There exists {r n } n, 0 <r n < 1, such that r n 1 and N u i (z n )f n,rn (ϕ i (z n )) >ɛfor all n. Since f n,r n 0 uniformly on compact subset of D, it follows that N u i C ϕi is not compact on A(D).

5 Weighted composition operators 1949 The following theorem gives a necessary condition for compactness of infinite sum of weighted composition operators on disk algebra. n N ({z o},k) Theorem 2.3. Let {ϕ n } be distinct functions in A(D) with ϕ n =1for each n 1 and {u n } a sequence in A(D) with u n <. If u n C ϕn is compact on A(D) then u n (z o )=0for every z o Z and for every k N({z o }), where Z={z T : ϕ n (z) =1for some n}, N ({z o })= {n 1: ϕ n (z o ) =1} and N ({z o },k)={m N({z o }):ϕ k (z o )=ϕ m (z o )}. Proof. Suppose u n C ϕn is compact on A(D). We claim that n N ({z o},k) u n (z o )= 0 for every z o Z and for every k N({z o }). Let z o ( Z and k N({z ) o }). m z + ϕk (z o ) For each positive integer m define f m as f m (z) =. Then 2ϕ k (z o ) f m A(D), f m = 1 and f m 0 uniformly on every compact subset of D. Since u n C ϕn is compact on A(D), there exists a subsequence {f mi } ( ) and a function F in A(D) such that u n C ϕn f mi F in A(D). That is u n f mi (ϕ n (z)) F (z) uniformly for z D. But f mi (ϕ n (z)) 0 for z < 1 and u n is uniformly bounded. So F (z) =0onD. However, as ( F is continuous on D, therefore F (z) =0on D. ) Hence u n C ϕn f mi 0

6 i 1950 H. Chandra and B. Singh uniformly on D. In particular, lim u n (z o ) f mi (ϕ n (z o )) = 0. Now interchanging limits, which is permissible under the given conditions, we get 0 = lim u n (z o ) C ϕn f mi (z o ) = lim u n (z o ) f mi (ϕ n (z o )) i i Thus u n C ϕn = lim n lim i = lim n lim i = lim n = n u j (z o ) f mi (ϕ j (z o )) j=1 n ( ϕj (z o )+ϕ k (z o ) u j (z o ) 2ϕ k (z o ) j=1 n j=1 j N ({zo},k) j N ({zo},k) u n (z o ). is compact on A(D) implies that every z o Z and for every k N({z o }). u j (z o ) n N ({z o},k) ) mi u n (z o ) = 0 for The converse of the above theorem is not true and we give the following example to show this. Example 2.1. Let ϕ 1 (z) = z+1,ϕ 2 2(z) = z+2,u 3 1(z) = 1 and u 2 (z) = 1. Then Z = {1}, N ({1}) ={1, 2}, N ({1}, 1) = {1, 2} and u n (1) = n N ({1},1) 0. But u 1 C ϕ1 + u 2 C ϕ2 is not compact on H. Therefore by Theorem 2.2, u 1 C ϕ1 + u 2 C ϕ2 is not compact on A(D). The following corollaries are simple consequences of Theorem 2.3. Corollary 2.1. Let ϕ 1,ϕ 2,ϕ 3,...,ϕ K (K 2) be distinct functions in A(D) with ϕ n = 1 and u 1,u 2,u 3,...,u K in A(D) with u n 0 for every n, 1 n K. If there exists a point z o Z such that u n (z o ) 0 for every subset n J Jof{1, 2,...,K}, then K u n C ϕn is not compact on A(D). Corollary 2.2. Let ϕ 1,ϕ 2,...,ϕ K (K 2) be distinct functions in A(D) with ϕ n = 1 and λ 1,λ 2,λ 3,...,λ K C with λ n 0 for every n, 1 n K. If λ n 0 for every subset J of {1, 2,...,K}, then K λ n C ϕn is not compact n J on A(D).

7 Norm 1951 Thus the sum K C ϕn is never compact on A(D) provided ϕ n = 1 for each n, 1 n K. An example of the non-compact operator of the form u n C ϕn on A(D) is the following. Example 2.2. Let ϕ n (z) = z+( 1)n 1 n.then Z = {1, 1}, N ({1}) ={1, 3, 5,...}, n+1 N ({ 1}) ={2, 4, 6,...},N ({1},k)={1, 3, 5,...} for every k N({1}) and N ({ 1},l)={2, 4, 6,...} for every l N({ 1}). For n 1, define u n as follows: u 2n 1 (z) = z and u 2 n 2n (z) = z. 2 n Since u n ( 1) 0, hence by Theorem 2.3 the operator u n C ϕn is n N ({ 1},l) non-compact on A(D). 3. Norm In this section we estimate the norm of sum of weighted composition operators on A(D). Let ϕ 1,ϕ 2,...,ϕ K (K 2) be distinct functions in A(D) with ϕ n 1 and u 1,u 2,u 3,...,u K in A(D) with u n 0 for every n, 1 n K. Let B = {z T : u n = u n (z), for all n, 1 n K} and Z = {z T : ϕ n (z) =1, for some n}. K Case 1. If B = φ, then u n C ϕn < K u n. Proof. Let f A(D) with f 1. Then K u n C ϕn f = sup K u n (z) f (ϕ n (z)) z 1 K sup u n (z) z 1 = sup { u 1 (z) + u 2 (z) + + u K (z) }. z 1 Since u 1 (z) + u 2 (z) + + u K (z) is continuous, this implies there exists a point z o T such that sup { u 1 (z) + u 2 (z) + + u K (z) } = u 1 (z o ) + z 1 u 2 (z o ) + + u K (z o ). Now B = φ implies that exists a u n such that u n (z o ) < u n and also u m (z o ) u m for every m n.

8 1952 Harish Chandra and Bina Singh Let ɛ = un un(zo). This implies u 2 n (z o ) < u n ɛ. Hence sup { u 1 (z) + u 2 (z) + + u K (z) } = u 1 (z o ) + u 2 (z o ) + + u K (z o ) z 1 < u 1 + u u K ɛ. K Therefore u n C ϕn f < K u n ɛ for all f A(D) with f 1. K Thus we have sup u n C ϕn f K u n ɛ. f 1 Since ɛ>0, this implies K u n C ϕn < K u n. Thus we get the following lemma. Lemma 3.1. Let ϕ 1,ϕ 2,...,ϕ K (K 2) be distinct functions in A(D) with ϕ n 1 and u 1,u 2,u 3,...,u K A(D) with u n 0for every n, 1 n K. If K u n C ϕn = K u n, then there exists a point z o T such that u n = u n (z o ) for all n, 1 n K. Case 2. If B φ and there exists a z B such that un(z ) = um(z ) u n u m for all n and m. Then one can easily seen that K u n C ϕn = K u n Case 3. If B φ and for each z B, un(z ) um(z ) u n u m Then we have the following theorem : for some n and m. Theorem 3.1. let ϕ 1,ϕ 2,...,ϕ K (K 2) be distinct functions in A(D) with ϕ n 1 and u 1,u 2,u 3,...,u K A(D) with u n 0for every n, 1 n K. Suppose that for each z B, un(z ) u n um(z ) u m for some n and m. Then K u n C ϕn = K u n if and only if there exists a z o B Z satisfying ρ (ϕ n (z o ),ϕ m (z o )) = 1 for each n and m with un(zo) u n um(zo) u m. To prove this, we need a lemma due to Gorkin and Mortini, whose easy modification is as follows. see ([6], Proof of theorem 5). Lemma 3.2. Let ϕ 1,ϕ 2,...,ϕ K be distinct functions in A(D) with ϕ n 1. Let z o D with z o = 1. If min ρ (ϕ n (z o ),ϕ m (z o )) = 1, then for e iθ j T, 1 n m j K, there exists a sequence f k in ball A(D) such that f k (ϕ n (z o )) e iθ j as k for every n.

9 Norm 1953 K Proof of Theorem 3.1. Suppose u n C ϕn = K u n. Then by Lemma 3.1 K it follows that B φ. Further u n C ϕn = K u n implies there exists a sequence of function {f k } ball A(D) satisfying K K u n C ϕn f k u n as k. Consequently, there exists a sequence in {z p } D such that K K u n (z p ) f k (ϕ n (z p )) u n as k and p. Without any loss of generality, assume that z p z o. This gives K K u n (z o ) f k (ϕ n (z o )) (3.1) u n as k. Since u n (z o ) f k (ϕ n (z o )) u n (z o ) for every n and k, therefore we get K K K u n = lim u n (z o ) f k (ϕ n (z o )) k u n (z o ) K u n. Thus K u n = K u n (z o ). This implies z o B. Now by (3.1) (3.2) f k (ϕ n (z o )) u n (z o ) as k for every n. u n Let n and m with u n(z o ) u n u m(z o ). By Schwarz s lemma u m ρ (f k (ϕ n (z p )),f k (ϕ m (z p ))) ρ (ϕ n (z),ϕ m (z p )) for each p 1 and for all k 1. Hence lim ρ (f k (ϕ n (z p )),f k (ϕ m (z p ))) lim ρ (ϕ n (z),ϕ m (z p )), which implies p p ρ (f k (ϕ n (z 0 )),f k (ϕ m (z p ))) ρ (ϕ n (z),ϕ m (z 0 )) for all k 1. Now by equation (3.2) 1 = lim ρ (f k (ϕ n (z o )),f k (ϕ m (z o ))) k ρ (ϕ n (z o ),ϕ m (z o )) Hence ρ (ϕ n (z o ),ϕ m (z o )) = 1. This shows that either ϕ n (z o ) =1or ϕ m (z o ) = 1. Therefore z o Z. Conversely, suppose that there is a z o B Z satisfying ρ (ϕ n (z o ),ϕ m (z o ))=1

10 1954 Harish Chandra and Bina Singh for each n and m with u n(z o ) u n u m(z o ) u m. Let ρ (ϕ n (z o ),ϕ m (z o )) = γ n,m.on the set I = {1, 2,...,K}, we define an eqivalence relation as follows. For n, m I, n m if and only if γ n,m < 1. By given condition, it follows that, if n m then u n(z o ) u n = u m(z o ). Now we choose one element from each equivalence class and let J denote the collection of all such elements of I. Note that u m for every n, m J with n m, we have ρ (ϕ n (z o ),ϕ m (z o )) = 1. By Lemma 3.2, there exists a sequence {f k } in ball A(D) satisfying (3.3) f k (ϕ n (z o )) u n (z o ) as k forevery n J. u n K K Now u n C ϕn lim u k n (z o ) f k (ϕ n (z o )) = lim k u m (z o ) f k (ϕ m (z o )) + u n (z o ) f k (ϕ n (z o )) m J n I\J. Using equation (3.3), we get u m (z o ) f k (ϕ m (z o )) u m as k. m J m J Also for each n I\J there is m J with n m such that ρ (ϕ n (z o ),ϕ m (z o )) < 1. Consequently (3.4) u n (z o ) u n = u m(z o ) u m. Now using equations (3.2) and (3.4) it follows that (3.5) f k (ϕ n (z o )) u m (z o ) u m as k. Combining equations (3.4) and (3.5), we get u n (z o ) f k (ϕ n (z o )) u m (z o ) u m u n (z o ) u m (z o ) u m = u n (z o ) as k. Thus we get K u n C ϕn K u n. K Therefore u n C ϕn = K u n. This completes the proof. Theorem 3.2. Let {ϕ n } be distinct functions in A(D) with ϕ n 1 for each n 1 and {u n } a sequence in A(D) with u n <.Suppose that for each z B, un(z ) um(z ) u n u m for some n and m. Then u n C ϕn =

11 Norm 1955 u n if and only if there exists a z o B Z satisfying ρ (ϕ n (z o ),ϕ m (z o )) = 1 for each n and m with un(zo) u n um(zo) u m. N Proof. For N 2, by Theorem 3.1, we get u n C ϕn = N u n for each N 1. Hence N u n C ϕn = lim u n C ϕn N = lim N N u n = u n. Example 3.1. Let ϕ n (z) = z +( 1)n 1 n. For n 1, we define u n as follows: n +1 u 2n 1 (z) = z and u 2 n 2n (z) = z. 2 n Then B = {1, 1} and Z = {1, 1}. Let m be even and n be odd. Then we have ρ (ϕ n (z o ),ϕ m (z o )) = 1 and u n(z o ) u n u m(z o ) u m for every z o B. Hence by Theorem 3.2. u n C ϕn = u n = =2 References [1] M.D. Contreras and A.G. Hernandez-Diaz, Weighted composition operators in weighted Banach spaces of analytic functions, Journal of Australian Mathematical Society (Series A), 69 : 41-60, [2] M.D. Contreras and A.G. Hernandez-Diaz, Weighted composition operators on Hardy spaces, Journal of Mathematical Analysis and Applications, 263 : , [3] M.D. Contreras and S. Diaz-Madrigal, Compact type operators defined on H, Contemp. Math., 232 : , [4] C.C. Cowen, B.D. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC Press Boca Raton, [5] J.B. Garnett, Bounded Analytic Functions, Academic Press, New York, [6] P. Gorkin and R. Mortini, Norms and essential norms of linear combinations of endomorphisms, Transactions of American Mathematical Society, 358 : , [7] K. Hoffman, Banach Spaces of Analytic Functions, Prentice Hall, Englewood cliffs, NJ [8] T. Hosokawa and K. Izuchi, Essential norms of differences of composition perators on H, Journal of Mathematical Society Of Japan, 57 : , 2005.

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