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1 EXTREME POINTS OF THE CLOSED CONVEX HULL OF COMPOSITION OPERATORS TAKUYA HOSOKAW A Abstract. W e study the extreme points of the closed convex hull of the set of all composition operators on the space of bounded analytic functions and the disk algebra.. Introduction Let D be the open unit disk. W e denote by D its closure and its boundary. Let H(D) be the set of all analytic functions on D and S(D) be the set of all analytic self-maps of D. Every analytic self-map ' 2 S(D) induces the composition operator C ' on H(D) dened by C ' f(z) = f('(z)): Let H be the set of all bounded analytic functions on D. Then H is a Banach algebra with the supremum norm kfk = sup jf(z)j: z2d Every composition operator is bounded on H and kc ' k =. It is known that C ' is compact on H if and only if k'k <. Let X be an analytic functional Banach space on D, that is, each element is analytic on D and the evaluation at each point of D is a non-zero bounded linear functional on X. Let C(X ) be the collection of all bounded composition operators on X, endowed with the operator norm topology. Then C(X ) is a semigroup with respect to the products, but is not a linear space. In the case that X is the Hilbert Hardy space H 2, Shapiro and Sundberg [] raised the following three problems: (i) Characterize the path components of C(H 2 ). (ii) W hich composition operators are isolated in C(H 2 )? (iii) W hich di erences of composition operators are compact on H 2? Though many mathematicians have tried to solve these problems and gotten several results, all of these three problems are still open. Date: Received date / Revised version date. 99 Mathematics Subject Classication. 47B33. Key words and phrases. composition operator, extreme point. This work is supported by the Korean Research Foundation Grant funded by Korean Government(MOEHRD, Basic Research Promotion Fund)(KRF-25-7-C7).

2 2 HOSOKAW A In [9], MacCluer, Ohno and Zhao solved (i) and (ii) of the problems above in the setting of C(H ). Their results were described in terms of the pseudo-hyperbolic distance on D. For p 2 D, let p be the automorphism of D exchanging for p. Then p has the following form; p (z) = p z pz : The pseudo-hyperbolic distance ½(z; w) between z and w in D is dened by ½(z; w) = j z (w)j = z w zw : Here we dene the induced distance d ½ on S(D), that is, d ½ ('; Ã) = sup ½('(z); Ã(z)) z2d for ' and Ã in S(D). In [9] the operator norms of the di erences of composition operators on H are estimated as following; kc ' C Ã k = 2 2p d ½ ('; Ã) 2 : d ½ ('; Ã) Hence C(H ) can be identied with the space S(D; d ½ ). W e denote C '» X C Ã if they are in the same component of C(X ). In [9], it is proved that C '» H C Ã if and only if d ½ ('; Ã) <. Let Y be a convex subset of a locally convex space. W e recall that an element y of Y is called an extreme point of Y if y can not be written as y = ( r)y + ry 2 with < r < and distinct elements y ; y 2 2 Y. For a normed space Z, we denote by U Z the closed unit ball of Z. By de Leeuw- Rudin's Theorem (see [4] or [6, Chapter 9]), ' is an extreme point of U H if and only if () Z 2¼ log( j'(e iµ )j)dµ = : MacCluer, Ohno and Zhao proved that if C ' is isolated in C(H ), then ' is an extreme point of U H. In [7], the converse was proved. The connected components of C(H ) are characterized by the equivalence relation which is in the similar form of the Gleason parts of the maximal ideal space of H. In this sense, the isolated point of C(H ) corresponds to the single Gleason part. W e recall the notion of exposed points of U Z. An element x of U Z is called an exposed point of U Z if there exists a linear functional L in Z such that L(x) = klk Z = and Re L(y) < for any y 2 U Z, y 6= x. It is easy to see that an exposed point of U Z is an extreme point of U Z. Amar and Lederer proved that ' is an exposed points of U H if and only if je(')j >, where E(') is the set of all ³ such that j'(³)j = and je(')j is the Lebesgue measure of E(') (see [] and [5, p.23]). On the other hand, from the fact that C(X ) is not a linear space we derive the problem of when the linear combination of composition operators is a composition operator. In section 2, we consider this problem for H.

3 EXTREME POINTS OF COMPOSITION OPERATORS 3 This problem for the case of the nite linear combinations was given as an exercise in [3, Chapter ], which concludes that there is no nite linear combination of composition operators which is a composition operator. Next we study this problem for the case of the innite linear combinations. To do this, we prepare some notations. Denote by hc(x )i the collection of all nite linear combinations of composition operators on X and denote by hc(x )i + the collection of all nite linear combinations of composition operators on X with positive coe±cients. Moreover, let L(X ) and L + (X ) denote the operator norm closure of hc(x )i and hc(x )i +, respectively. W e remark that the closed convex hull of C(H ) is the closed unit L+(H ) (see Proposition 2.5). W e consider the problem of which composition operators are the extreme points of U L+ (H ). The main theorem of this paper (Theorem 2.9) shows that the extremeness of C ' in U L(H ) requires the extremeness of ' of U H. This result gives the answer for the innite linear combination problem, that is, there is the innite linear combination of composition operators which is a composition operator. W e also show that the exposed points of U H induce the extreme composition operators in U L+(H ). In section 3, we consider the case of another function space, the disk algebra A. W e will study the set C(A) and can obtain the complete answer when C ' is an extreme point of L(A). 2. Extremeness of composition operators on H At rst, we observe that composition operators are linearly independent in hc(h )i. Proposition 2.. Let ' ; ; ' n be distinct analytic self-maps and let ; ; n 2 C. If C ' + + nc 'n is the zero operator on H, then = = n =. Proof. Put T = C ' + + nc 'n and f m (z) = z m. By the Identity Theorem, there is a subset E of D with zero measure such that ' j 6= ' k on D n E for all j; k such that j 6= k. Fix a point p 2 D n E. Since T f m (p) = for any m = ; ; ; n, we have that ' (p) ' 2 (p) ' n (p) 2 B C B A = B A : ' (p) n ' 2 (p) n ' n (p) n By the denition of p and Vandermonde's determination, ' (p) ' 2 (p) ' n (p) = Y ('... k (p) ' j (p)) 6= : ' (p) n ' 2 (p) n ' n (p) n j<k Since this matrix is invertible, we have that = = n =. n

4 W 4 HOSOKAW A Hence we conclude that there is no nite linear combination of composition operators which is a composition operator. Let H be the convex hull of fc ' ; ; C 'n g. For T 2 H, put T = kc 'k : k= Then we have that k for any k = ; ; n and P n k= k =. Moreover kt k = T = k = : W e determine the extreme points of H. Corollary 2.2. Let ' ; ; ' n be distinct analytic self-maps and H be the convex hull of C ' ; ; C 'n. Then the set of all extreme points of H is the vertexes fc ' ; ; C 'n g. W W Proof. By Proposition 2., we have that each C 'k is an extreme point of H. e suppose that T 2 H n fc ' ; ; C 'n g. ithout loss of generality, we assume that > and put S = kc 'k : Then we have that ksk = and T = C ' + ( )S. This means that T is not an extreme point of H. Next, let F = hc ' ; ; C 'n i be the nitely generated vector space over C. W e consider the extreme points of U F. It is easy to see that (2) kc 'k j kj: k= e can also see that the equality of (2) holds if =j j = = n=j nj. Recently, Izuchi and Ohno characterized in [8] when the equality of (2) holds. Theorem 2.3. [Izuchi-Ohno, [8]] Let ' ; ; ' n be distinct analytic selfmaps and ; ; n be nonzero complex numbers. Suppose that k=j kj 6= k=j kj for some k 6= k. Then (3) kc 'k = j kj k= if and only if there exists a sequence fz j g in D such that ½(' k (z j ); ' k (z j ))! as j! for every k; k with k=j kj 6= k=j kj. If the equality (3) holds for any linear combination in F, we can determine the extreme points of U F by the similar proof of Corollary 2.2. k= k=2 k= k=

5 W EXTREME POINTS OF COMPOSITION OPERATORS 5 Corollary 2.4. Let ' ; ; ' n be distinct analytic self-maps and F be the vector space spanned by C ' ; ; C 'n over C. Suppose that there exists a sequence fz j g in D such that ½(' k (z j ); ' k (z j ))! as j! for every k; k with k=j kj 6= k=j kj. Then the extreme points of U F are in the form of e iµ C 'k for µ 2 [; 2¼) and k = ; ; n. To investigate the innite linear combinations of composition operators, we give the estimation of the operator norm of the element of L + (H ) in the following proposition. Proposition 2.5. If T is in L + (H ), then kt k = T. Proof. Let ft j g ½ hc(h )i + be the Cauchy sequence which converges to T 2 L + (H ). Put n j X T j = where j;k >. Then we have that k= j;k C 'j;k n j X kt j k = T j = for each j. Since kt T j k kt T j k!, we have that T is a constant function whose value is the limit of T j. The assertion follows from kt k kt j k kt T j k! : k= j;k Here we will construct some example of elements of L + (H ) induced by the continuous curve fc 't g t2[;] in C(H ). W e dene that T n = k= n C ' kn : Then Proposition 2.5 implies that kt n k =. For f 2 H and p 2 D, we have that Z T n f(p) = n f(' k (p))! f(' s (p))ds n k= as n!. Since ft n fg is a Cauchy sequence in H, we have that Z f(' s (z))ds 2 H : e denote by I f'tg the following integral operator: I f'tgf(z) = Z f(' s (z))ds: Then the Banach-Steinhaus Theorem implies the following lemma.

6 W 6 HOSOKAW A Lemma 2.6. If fc 't g t2[;] is a continuous curve in C(H ), then the corresponding integral operator I f'tg is in U L+(H ). Example 2.7. (i) Suppose that C '» H C Ã. Put ' t = ( t)' + tã. Then fc 't g t2[;] is a continuous curve in C(H ) (see Lemma 5 and Lemma 6 of [9]) and I f'tgf(z) = ( F (Ã(z)) F ('(z)) Ã(z) '(z) f ('(z)) if '(z) 6= Ã(z) if '(z) = Ã(z) where F (z) is the primitive function of f(z). (ii) Suppose that k'k <. Choose a positive number r such that r < k'k. W e dene that ' t (z) = '(z)+re 2¼it z. Then k' t k < and ' t 2 S(D) for all t. Since every ' t (D) is a relatively compact subset of D, d ½ (' s ; ' t )! as s! t. Thus fc 't g t2[;] is a closed continuous curve in C(H ). By the Cauchy's Formula, we have that I f'tg = C '. e remark that the condition k'k < induces that C ' is not an extreme point of U L+ (H ). Proposition 2.8. If C ' is compact on H, then C ' is not an extreme point of U L+ (H ). Proof. Let f 2 H and p 2 D. From (ii) of Example 2.7, we have that C ' f(p) = Z 2 f '(p) + rp e 2¼is ds + Z 2 f '(p) + rp e 2¼is ds: Let ¾ t (z) = '(z) + re ¼it z and t (z) = '(z) re ¼it z. By changing variables, (4) C ' = 2 I f¾ tg + 2 I f tg: It is easy to see that I f¾tgz = '(z) 2rz and I ¼i f tgz = '(z) + 2rz ¼i : Since I f¾tg 6= I f tg, we can conclude that C ' is not an extreme point. The proposition above can be applied to more general cases. The following theorem gives a necessary condition for C ' to be an extreme point of U L+(H ). Theorem 2.9. If C ' is an extreme point of U L+ (H ), then ' is an extreme point of U H. Proof. Suppose that ' is not an extreme point, then the de Leeuw-Rudin's Theorem implies that Z 2¼ log( j'(e iµ )j)dµ > :

7 W Put (5) EXTREME POINTS OF COMPOSITION OPERATORS 7 µ Z 2¼ e iµ + z!(z) = exp 2¼ e iµ z log( j'(eiµ )j)dµ : Then! is an outer function in H and j'j + j!j on D. Here we x a positive number r 2 (; ) and dene that ' t = '+re 2¼it!. W e will see that fc 't g is a closed continuous curve in C(H ). To see this, it is su±cient that d ½ (' ; ' t )! as t!. Let D = fz 2 D : j'(z)j < rg and D 2 = D n D. For j = ; 2, we put d j ('; Ã) = sup z2d j ½('(z); Ã(z)): Since D is a compact subset of D, it is easy to check that d (' ; ' t )! as t!. On the other hand, using j'(z)j r on D 2 and j'j j!j, r!(z)( e 2¼it ) d 2 (' ; ' t ) = sup ('(z) + r!(z))('(z) + re 2¼it!(z)) z2d 2 rj!(z)jj e 2¼it j sup z2d 2 j'(z)j 2 2rj'(z)jj!(z)j r 2 j!(z)j 2 rj e 2¼it j sup z2d 2 + j'(z)j 2rj'(z)j r 2 ( j'j) rj e 2¼it j = sup z2d 2 ( r 2 ) + ( r) 2 j'(z)j rj e2¼it j r 2! as t!. Hence we get d ½ (' ; ' t )! as t! and this implies that fc 't g is continuous. By the Cauchy's Formula, we have that I f't g = C '. Hence, by the similar way to the proof of Proposition 2.8, we can show that C ' has the form (4), and so is not an extreme point. This concludes the proof. e remark that if ' is not an extreme point of U H, then C ' is not an extreme point of U L+(H ), and so is not an extreme point of U L(H ). Next we give a su±cient condition for C ' to be an extreme point of the closed convex hull U L+(H ) of C(H ). For a xed ' 2 S(D), let L ' be the norm closure of hc(h )nc ' i +. Then L ' ½ L + (H ). If ' is not an extreme point of U H, Theorem 2.9 implies that L ' = L + (H ). W e estimate the distance to C ' induced by an exposed point ' of U H from L '. Lemma 2.. Let ' 2 S(D) be an exposed point of U H. Then kc ' T k = + T for any T 2 L '. Proof. First, we suppose that T 2 hc(h ) n C ' i +. Put T = P n k= kc 'k with each k >. Then we have that kt k = T = P n k= k. Since ' k 6= '

8 8 HOSOKAW A and je(')j >, there exists a sequence fz j g ½ D such that '(z j )! and each ' k (z j )! k 2 D n f g as j!. Let µ z + m f m (z) = : 2 Then f m ( ) = and jf m (z)j < for z 2 D n f g. Let g r (z) = r z rz : for r 2 (; ). Then g r is an automorphism of D such that g r () = and g r () = r. Given " >, we put r = "=4. There exists a positive integer N and a constant number ± > such that for all k and for all j > N, j' k (z j ) j ±. Then there exists a positive integer M such that for each k and for all j > N, (6) jf M (' k (z j ))j 3 : For p 2 D with jpj =3, (7) jg r (p) j = ( r)( + p) rp " 4 + jpj rjpj " 2 : Here we can choose a positive integer N 2 such that for all j > N 2, (8) jg r ± f M ('(z j )) j "=2: Put N = maxfn ; N 2 g. Since g r ± f M is a unit vector of H, we have that for j > N, kc ' T k (C ' T )(g r ± f M )(z j ) = g r ± f M ('(z j )) k g r ± f M (' k (z j )) ³ + k= k By (6), (7), and (8), we get k= g r ± f M ('(z j )) k gr ± f M (' k (z j )) : k= kc ' T k + T " 2 " 2 = + T ": Since " is arbitrary, we have that kc ' T k = + T for any T 2 hc(h ) n C ' i +. Next, x T 2 L '. Given " >, there exists T 2 hc(h ) n C ' i + such that kt T k ". Since jt T j ", we have that T T ". Thus we obtain that kc ' T k kc ' T k kt T k + T kt T k + T 2":

9 This concludes the proof. EXTREME POINTS OF COMPOSITION OPERATORS 9 Here we give the following theorem. Theorem 2.. Let ' be in S(D). If ' is an exposed point of U H, then C ' is an extreme point of U L+ (H ). Proof. Let ' be an exposed point of U H. To prove C ' is an extreme point of U L+(H ), suppose not. Then there exist two distinct operators S, T in U L+ (H ), and r 2 (; ) such that ksk = kt k = and C ' = ( r)s + rt: There exist a; b 2 [; ] and two operators S, T in L ' such that S = a C ' + S and T = b C ' + T. Then we have that ( ( r)a rb) C ' (( r)s + rt ) = : If ( r)a rb 6=, then (9) C ' ( r)s + rt ( r)a rb = : On the other hand, since S = S ac ' = a and T = b, Lemma 2. implies that C ' ( r)s + rt ( r)( a) + r( b) ( r)a rb = + = 2: ( r)a rb This contradicts (9). Hence ( r)a + rb = and ( r)s + rt =. Since the point is an extreme point of the interval [; ], we have that a = b =. Then we obtain that S = C ' + S and S 6=. Proposition 2.5 implies that ksk = S = + S = + ks k > : This is a contradiction. Now we conclude that C ' is an extreme point of U L+ (H ). 3. The case of the disk algebra In this section, we consider composition operators on the disk algebra A. Recall that the disk algebra A is the set of all continuous functions on D which are analytic on D. Then A is a Banach algebra with the norm sup jf(z)j: z2d By the maximum modulus principle, we can see that this norm is equal to kfk. To dene C ' on A, we need the condition C ' z = ' 2 A. Denote by S(D) the set of functions ' which is analytic on D and continuous on D such that '(D) ½ D. Then S(D) is the closed unit ball U A of A and each ' 2 S(D) induces C ' which acts on A. Denote by kc ' k A the operator norm of C ' on A. If ' is a constant function with value then ' is not in S(D) but in S(D). Dene that = f' By the maximum modulus principle,

10 HOSOKAW A it is shown that S(D)n = S(D)\A. W e remark that ' is an extreme point of S(D) if and only if ' 2 A and the condition () holds (see [6, p. 39]). Moreover, just as the case of H, it is well-known that kc ' k A = for every ' 2 S(D) and C ' is compact on A if and only if k'k < or ' 2. W e denote that K = fc ' is compact on Ag and T = fc ' : ' 2 g. Then we have T ½ K. Here we extend the pseudo-hyperbolic distance to D, denote by ½, as following. For z; w 2 D, ½(z; w) = ½(z; w). For z ½ if w = z ½(z; w) = if w 6= z : Of course, this extension is not continuous. dened on S(D), that is, The induced distance d ½ is d ½ ('; Ã) = sup ½('(z); Ã(z)): z2d Now we can obtain the following results on the topological structure of C(A) by a similar proof for C(H ) given in [7] and [9]. Theorem 3.. Let '; Ã be in S(D). (i) kc ' C Ã k A = 2 2p d ½ ('; Ã) 2. d ½ ('; Ã) (ii) C '» A C Ã if and only if kc ' C Ã k A < 2. (iii) The following are equivalent. (a) C ' is isolated in C(A). (b) For all C Ã 6= C ', kc ' C Ã k A = 2. (c) ' is an extreme point of S(D). (d) Z 2¼ log( j'(e iµ )j)dµ =. (iv) Every C ' 2 T is compact on A and is isolated in C(A). (v) K n T itself is a path component of C(A). Proof. (i) Let '; Ã 2 S(D) and ' 6= Ã. First, suppose C ' and C Ã are in C(A) n T. For z, w 2 D, the induced distance d A (z; w) is dened by d A (z; w) = supfjf(z) f(w)j : f 2 A; kf k < g: In [], Madigan estimated d A (z; w) precisely as below. d A (z; w) = 2 2p ½(z; w) 2 : ½(z; w)

11 EXTREME POINTS OF COMPOSITION OPERATORS Since (2 2 p x 2 )=x is continuous and increasing on [; ], we have that kc ' C Ã k A = sup sup f('(z)) f(ã(z)) f 2S(D) z2d = sup sup f('(z)) f(ã(z)) z2d f 2S(D) = sup d A (z; w) z2d = 2 2p d ½ ('; Ã) 2 : d ½ ('; Ã) Next we suppose C ' 2 T and C Ã 62 T. This means that ' e iµ. Then d ½ ('; Ã) =. To prove (i), we will show that kc ' C Ã k A = 2. Indeed, there exists p 2 D such that Ã(p) 2 D. Recall that p (z) = (p z)=( pz). Put q = Ã(p) (e iµ ) and f(z) = q Ã(p) (z). Let fr n g be a sequence of real numbers increasing to in [; ) and f n (z) = rn ±f(z). Then f n is an automorphism and in S(D). Now we get kc ' C Ã k A k(c ' C Ã )f n k jf n ± '(p) f n ± Ã(p)j = j rn () rn ()j = j + r n j! 2; as n!. Conversely, since kc ' k A = kc Ã k A =, we have that kc ' C Ã k A 2. W e conclude that kc ' C Ã k A = 2. Suppose that C ' and C Ã are in T, that is, ' e iµ and Ã e i³. Then there exists f 2 A, kf k = such that f(e iµ ) = and f(e i³ ) =. Hence we have that kc ' C Ã k A k(c ' C Ã )fk = 2: The proof of (i) is nished. (ii) The same proof of [9, Theorem ] can be applied. (iii) The equivalence of (a) and (b) follows from (ii). The equivalence of (c) and (d) was mentioned in [2] and [6]. The same proof of [7, Theorem 4..] shows that (d) implies (a). Though Chandra showed that (a) implies (d) in [2], we will give the simpler proof. Suppose that log( j'(e iµ )j) is integrable. Then the outer function! of (5) is in A and j!(z)j j'(z)j for z 2 D. Put ' t = ' + t! for jtj <. If j'(z)j =, then!(z) = and ½('(z); ' t (z)) =. If j'(z)j <, then we have '(z) ' t (z) jtj ½('(z); ' t (z)) = '(z)' t (z) = jtj jtj < j j'(z)j2!(z) t'(z)j for jtj < =2. Thus d ½ ('; ' t ) <. Hence we have C '» A C 't. This means that C ' is not isolated. Now the proof of (iii) was completed.

12 2 HOSOKAW A (iv) For C ' 2 T, C ' (A) = C is the subspace of dimension one. Hence C ' is compact on A. Since j'j = on D, (iii) implies that C ' is an isolated point of C(A). (v) At rst, we will show that K n T is path connected in C(A). Let C ' 2 K n T. For r 2 [; ], C r' 2 K n T and kr'k k'k <. d ½ (r'; s') = jr sj j'(z)j sup z2d rsj'(z)j 2 k'k k'k 2 jr sj! as r! s. (i) implies that the curve fc r' g r2[;] is continuous in C(A). Then we have that C '» A C for every C ' 2 K n T. Thus we conclude that C '» A C Ã for C ' ; C Ã 2 K n T. This means that K n T is path connected. Next we will prove that K n T forms a component. For C ' 2 K n T and C Ã 2 C(A) n (K n T ). Then there exists a point p such that j'(p)j < and jã(p)j =. Then we have d ½ ('; Ã) =. Since kc ' C Ã k A = 2, (ii) implies that C ' 6» A C Ã. This completes our proof. Denote by Comp X (') the path component of C(X ) which contains C '. By Lemmas 5 and 6 of [9], if C '» H C Ã (C '» A C Ã, respectively), then the \segment" ' t = ( t)'+tã induces a continuous path fc 't g in C(H ) (C(A), respectively). From this fact, we can immediately get the following corollary, which mentions the relation between the topological structure of C(A) and that of C(H ). Corollary 3.2. Let C ' and C Ã be in C(A)nT. Then we have the following. (i) Comp A (') = Comp H (') \ C(A). (ii) C '» A C Ã if and only if C '» H C Ã. (iii) C ' is isolated in C(A) if and only if C ' is isolated in C(H ). Now we can prove the converse of Theorem 2.9 on the setting of L(A). Theorem 3.3. Let C ' be in C(A). Then the followings are equivalent: (i) C ' is an extreme point of U L(A). (ii) C ' is an isolated point of C(A). (iii) ' is an extreme point of S(D). Proof. By Theorem 3., it is enough to prove the equivalence of (i) and (iii). Since we can prove the implication (i) ) (iii) by the same proof of Theorem 2.9, we need only to prove the implication (iii) ) (i). Suppose that ' is an extreme point of S(D) and C ' = ( r)s + rt for S and T in U L(A). Let Sz k = ¾ k and T z k = k. Then ¾ k and k are in the closed unit ball of A and ' k = ( r)¾ k + r k. W e remark that ' is an extreme point of S(D) if and only if ' k is an extreme point of S(D) for any positive integer k. Thus we get ¾ k = k = ' k for any positive integer k. Since the polynomials are dense in A, we get S = T = C '. This concludes the proof.

13 W EXTREME POINTS OF COMPOSITION OPERATORS 3 e present two problems. Problem. (i) The proof of Theorem 3.3 depends on the density of polynomials in A. However the density is not true in the case of H. Does Theorem 3.3 hold on the case of H? (ii) Theorem 3.3 does not answer the problem of whether the other extreme points exist or not. More precisely, we can ask the problem: whether the extreme point T of U L(A) is a composition operator or not? Acknowledgement. The author would like to thank the referee for many valuable suggestions and detailed criticisms on this paper. References [] E. Amar and A. Lederer, Points expos es de la boule unit e de H (D), C. R. Acad. Sci. Paris S er. A. 272 (97), 449{452. [2] H. Chandra, Isolation amongst composition operators on the disc algebra, J. Indian Math. Soc.(N.S.) 67 (2), 43{52. [3] C.C. Cowen and B.D. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Raton, 995. [4] K. deleeuw and W. Rudin, Extreme points and extreme problems in H, Pacic J. Math. 8 (958), 467{485. [5] J. B. Garnett, Bounded Analytic Functions (revised rst edition), Springer-Verlag, 26. [6] K. Ho man, Banach Spaces of Analytic Functions, Prentice Hall, Englewood Cli s, N. J., 962. [7] T. Hosokawa, K. Izuchi and D. Zheng, Isolated points and essential components of composition operators on H, Proc. Amer. Math. Soc. 3 (22), 765{773 [8] K. Izuchi and S. Ohno, Linear combinations of composition operators on H, J. Math. Anal. Appl. 338 (28), 82{839. [9] B. MacCluer, S. Ohno and R. Zhao, Topological structure of the space of composition operators on H, Integral Equation Operator Theory, 4 (2), 48{494. [] K. Madigan, Composition operators into Lipschitz type spaces, Thesis, SUNY Albany, 993. [] J. Shapiro and C. Sundberg, Isolation amongst the composition operators, Pacic J. Math. 45 (99), 7{52. Institute of Basic Science, Andong National University, Andong Korea address: turtlemumu@yahoo.co.jp

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

Int. Journal of Math. Analysis, Vol. 4, 2010, no. 39, 1945-1956 Compactness and Norm of the Sum of Weighted Composition Operators on A(D) Harish Chandra and Bina Singh Department of Mathematics and DST-CIMS