BOLLETTINO UNIONE MATEMATICA ITALIANA

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1 BOLLETTINO UNIONE MATEMATICA ITALIANA B. Yousefi, S. Jahedi Composition operators on Banach spaces of formal power series Bollettino dell Unione Matematica Italiana, Serie 8, Vol. 6-B (2003), n.2, p Unione Matematica Italiana < L utilizzo e la stampa di uesto documento digitale è consentito liberamente per motivi di ricerca e studio. Non è consentito l utilizzo dello stesso per motivi commerciali. Tutte le copie di uesto documento devono riportare uesto avvertimento. Articolo digitalizzato nel uadro del programma bdim (Biblioteca Digitale Italiana di Matematica) SIMAI & UMI

2 Bollettino dell Unione Matematica Italiana, Unione Matematica Italiana, 2003.

3 Bollettino U. M. I. (8) 6-B (2003), Composition Operators on Banach Spaces of Formal Power Series. B. YOUSEFI - S. JAHEDI dedicated to the memory of Karim Seddighi Sunto. Supponiamo che ]b(n)( sia una successione di numeri positivi e 1 GpE. Consideriamo lo spazio H p (b) di tutte le serie di potenze f(z) 4! f (n) z n, tali che! Nf (n)n p b(n) p E. Supponiamo che 1 1 1! 41 e n j 4 per un intero non-negativo j. Dimostriamo che se C f è compatto su H p (b), allora il limite p n41 b(n) non-tangenziale di f ( j11) ha modulo maggiore di uno, in ogni punto della frontiera del disco unitario aperto. Dimostriamo anche che se C f è di Fredholm su H p (b), allora W deve essere un automorfismo del disco unitario aperto. Summary. Let ]b(n)( be a seuence of positive numbers and 1 GpE. We consider the space H p (b) of all power series f(z) 4! f (n) z n such that! Nf (n)n p b(n) p 1 E. Suppose that and! n j 4 for some nonnegative integer j. We show that if C W is compact on H p (b), then the non-tangential p n41 b(n) limit of W ( j11) has modulus greater than one at each boundary point of the open unit disc. Also we show that if C W is Fredholm on H p (b), then W must be an automorphism of the open unit disc. Introduction. First in the following, we generalize the defintions coming in [5]. Let ]b(n)( be a seuence of positive numbers with b(0)41 and 1GpE. We consider the space of seuences f4]f (n)( such that V f V p 4V f V p b4! Nf (n)n p b(n) p E. The notation f(z) 4! f (n) z n shall be used whether or not the series converges for any value of z. These are called formal power series. Let H p (b) de- notes the space of such formal power series. These are reflexive Banach spaces with the norm VV b ([4]) and the dual of H p (b) is H (b p/ ) where 1 1 p

4 482 B. YOUSEFI - S. JAHEDI 1 41 and b p/ 4]b(n) p/ ( n ([6]). Also if g(z) 4! g (n) z n H (b p/ ), then VgV 4! Ng (n)n b(n) p. The Hardy, Bergman and Dirichlet spaces can be viewed in this way when p42 and respectively b(n) 41, b(n) 4 (n11) 21/2 and b(n) 4 (n11) 1/2 b(n11). If lim 41 or lim inf b(n) 1/n 41, then H p (b) consists of functions analytic on the open unit disc U. It is convenient and helpful n b(n) n to introduce the notation a f, gb to stand for g( f ) where fh p (b) and g H p (b)*. Note that af, gb 4! f (n) g (n)b(n) p. Let f (n) 4d (n). So f (z) 4z and then ] f ( is a basis such that V f V4b(). Clearly M z, the operator of multiplication by z on H p (b) shifts the basis ] f (. Remember that a complex number l is said to be a bounded point evaluation on H p (b) if the functional of point evaluation at l, e l, is bounded. The ( j functional of evaluation of the j-th derivative at l is denoted by e ) l. The function W in H p (b) that maps the unit disc U into itself induces a composition operator C W on H p (b) defined by C W f4f i W. The operator C W is Fredholm, if it is invertible modulo the compact operators. If C W is a bounded invertible operator, then W must be an automorphism of U, that is a one to one map of U onto U. We say an analytic self-map W of U has an angular derivative at w U, if for some h U the non-tangential limit of W(z)2h when zkw, exists and is z2w finite. We call this limit the angular derivative of W at w and denoted it by W8(w). Main results. We suppose that H p (b) consists of functions analytic on the open unit disc U. We study the Fredholm composition operator C W and investigate the compactness and essential norm of C W acting on the Banach space H p (b). LEMMA 1. Let X be a Banach space of analytic functions on a domain V in C. If there exists a seuence of functions g in the dual space X * such that Vg V41 and g K0 wealy with VC * W (g )VK0, then C W is not Fredholm on X. PROOF. Suppose S is any bounded operator on X *. Then by the hypothesis VSC W *(g )VGVSVVC W *( g )VK0 as K. Now let be an arbitrary compact operator on X *. Since is necessarily completely continuous, then we have V( g )VK0 ([2, p. 177, Proposition 3.3]). Thus V(I1) g VK1 for every compact operator on X *. This implies that SC W * 2I can not be compact, since else it should be V(I1 (SC W *2I))g VK1 that is a contradiction. Thus C W *, and hence C W, is not Fredholm. r

5 COMPOSITION OPERATORS ON BANACH SPACES ETC. 483 In the following we use the fact that e w H (b p/ ) and Ve w V 4! NwN n for all w in U ([6]). b(n) THEOREM 2. Let and! n j 4 for some non-negative integer j. If C W is Fredholm on H p (b), then W is an automorphism of the p b(n) disc. PROOF. It is well nown that if C W is Fredholm, then W is univalent since else the ernel of C W * will contain an infinite linearly independent set whose elements are differences of evaluation functionals. This is a contradiction, since dim er C W * E. So we need only show that W maps U onto U. If not, there exists v W(U)OU and z U such that W(z ) Kv. By the Open Mapping Theorem it should be Nz NK1. n Let j be the least non-negative integer such that the sum! j 4. If nf0 b(n) j40, set e 4 e z V e V. Then Ve V41. But z Nz lim Ve z V N! n 4 lim!nf0 b(n) 4 1 nf0 b(n) 4 and so if p is a polynomial in H p p(z (b), then lim ap, e b 4 lim ) 40. But polynomials are dense in H p (b), thus e K0 wealy as K. Since v is in U and V e z V W(z ) Kv, we have e W (z ) Ke v. Since we also have Ve z VK, we conclude that VC W * e z V4Ve W(z ) V/Ve z V tends to zero. So by Lemma 1, C W is not Fredholm that is a contradiction. If jd0, let e 4 e ( j) z where e ( j) V e z ( j) V z is the functional of evaluation of the j-th 1 derivative at z. Note that e w (z) 4! b(n) p wn z n and e w ( j) 4 d j e w. Thus dw j e z ( j) 4! Since Nz NK1 and! nf0 n(n21)(n22)r(n2j11) (z ) n2j n j 4, we have p b(n) b(n) p z n. lim V 4 lim (n(n21)r(n2j11)) Nz! N (n2j) 4. b(n) Since polynomials are dense in H p (b), by the same manner as in the previous case, we can see that e K0 wealy as K. Now we show that VC * W e VK0 as

6 484 B. YOUSEFI - S. JAHEDI K. A straightforward computation gives the following eualities: C * W e (1) z C * W e (2) z C * W e (3) z 4W8(z ) e (1) W(z ) 4W8(z ) 2 1e (2) W(z )1W9(z ) e (1) W(z ) 4W8(z ) 3 e (3) W(z)1WR(z ) e (1) W(z )12W9(z ) e (2) W(z )1W8(z ) W9(z ) e (2) W(z ) C * W e ( j) z 4W8(z ) j e ( j) W(z )1W ( j) (z ) e (1) W(z )1lower order terms where the lower order terms involves functionals of evaluation of derivatives of order less than j at W(z ) with coefficients involving products of derivatives of W at z of order less than j. From this it follows that VC * W e VK0 as K0. To see this first suppose that j41. Thus we have C * W e 4 W8(z ) e (1) W(z ) Ve z (1) V 4 aw, e b e (1) W(z ). But W(z ) Kn, where nu. So Ve W(z (1) ) VKVe n (1) VE. Also since e K0 wealy, aw, e b K0 as K. Thus indeed VC W * e VK0 as K. If jd1, remar that for all iej we have e W(z (i) )4!n41 n! (n2i)! (W(z )) n2i b(n) p and so Ve W(z (i) ) V 4!n4i (n(n21)r(n2i11) ) Nv(z )N n2i n i n ( j21) b(n) G! E, G!n4i b(n) n4i b(n) since j is the least non-negative integer such that! n j 4. Thus the limit b(n) of the norms of the functionals of evaluation of derivatives at W(z ) of order less than j remain bounded in U. Also, by the Principle of Uniform Boundedness Theorem sup Ve (i) z than j are bounded. Note that Ve ( j) z lim VC W * e V40 provided that K lim K VE for iej and all derivatives of W at z of order less 1 V V(W8(z ) ) j e ( j) VK and W(z ) KnU. Thus we have W(z )1W ( j) (z ) e W(z (1) ) V40.

7 COMPOSITION OPERATORS ON BANACH SPACES ETC. 485 Clearly ( ) 1 V V(W8(z ) ) j e ( j) W(z )1W ( j) (z ) e W(z (1) ) VG NW8(z )N j V Ve ( j) VWV j H p (b) Ve (1) z V W(z ) V1 NW ( j) (z )N Ve ( j) V V j z Ve (1) W(z ) VG Ve ( j) W(z ) V1NaW, e bn. Ve (1) W(z ) V. Note that VK and lim Ve z (1) VE, since 1 Ej. Also Ve W(z (i) ) VK Ve n (i) VE for i41, j and aw, e b K0, since e K0 wealy. Thus indeed the term in ( ) tends to zero as K and so VC W * e VK0 which by the lemma implies that C W is not Fredholm that is a contradiction. r Note that by the Julia Caratheodory Theorem ([3]), W has an angular derivative at w U if and only if W8 has non-tangential limit at w, and W has non-tangential limit of modulus one at w. Consider the open Euclidean disc, Julia disc, J(j, a) 4]zU; Nj2zN 2 Ea(12NzN 2 a )( of radius and center 11a j at, whose boundary is tangant to U at j. By the Julia s Lemma ([1]), if 11a j U and W is an analytic function such that B W 4 inf NW8(j)NE, then j U W(J(j, a) ) J(W(j), ab W ). Recall that the essential norm of C W is denoted by VC W V e and is the distance in the operator norm from C W to the compact operators. 41 and! n41 THEOREM 3. Let p integer j. Also for 0 GiGj let W (i) be an analytic self map of the unit disc U. If C W is a bounded operator on H p (b) and NW ( j11) (j)ng1 for some j U, then VC W V e F1 and C W is not compact. n j 41 for some non-negative b(n) PROOF. Let ]z ( be any seuence in U with z Kj. Also let j be the least non-negative integer such that the sum! n j 41. Set e 4 e ( j) z. Then n41 b(n) V e z ( j) V Ve V41 and by the same method used in the proof of Theorem 2, e K0 wealy as K. If K is any compact operator, then K * is completely continuous and since e K0 wealy, it should be VK * e VK0. By definition VC W V e 4 inf ]VC W 2 KV : K is compact( and VC W 2KV4V(C W 2K)* VFV(C W 2K)* e VFVC * W e V2VK * e V.

8 486 B. YOUSEFI - S. JAHEDI If K, then since VK * e VK0, we have VC W V e F lim that VC * W e V. Now we show Ve W(z ( j) ) V limvc W * e V4 lim V. Note that since NW ( j11) (j)ng1, by the Julia s Caratheodory theorem the nontangential limit of W (i) (j) have modulus one for i40, 1, R, j. If j40, then e 4 e z and C V e V W * e 4 e W(z). If j41, then e 4 e (1) z and z V e V z V e (1) z V C W * e 4W8(z ) e W(z) (1). But the non-tangential limit of W8(j) has modulus one V e z (1) V and so limvc W * e V4 limve W(z (1) ) V/Ve z (1) V. If jd1, then e 4e z ( j) /Ve ( j) V and z C W * e 4 1 V (W8(z ) j e ( j) W(z )1L j, ) where L j, is the sum of lower order terms and involves derivatives of order less than j at W(z ), i.e., terms of the type e W(z (i) ) (iej), with coefficients involving product of derivatives of W at z of order less than or eual to j. Remar that since j is the least non-negative integer such that! n j 41, then b(n) we have Ve (i) W(z ) V 4 (n(n21)r(n2i11)) NW(z )N n2i!n41 G b(n)!n41 n ( j21) b(n) E for iej. So lim Ve W(z (i) K ) V remains bounded for all i less than j. Also since VK and W (i) (j) has the non-tangential limit of modulus one for all igj, VL j, V thus indeed lim V e z ( j) V 40. Therefore lim VC * W e V4 lim NW8(z )N j Ve ( j) W(z ) V Ve W(z ( j) ) V 4 lim V. V V e ( j) W(z) V Now to complete h the proof it is sufficient to show that lim F1. For V e z ( j) V this set z 4g12 1 j. Then there exists a seuence ]r ( of non-negative numbers such that z is the point on J(j, r ) closest to 0. Therefore by the Ju-

9 COMPOSITION OPERATORS ON BANACH SPACES ETC. 487 lia s Lemma W(z ) W( J(j, r )) W(J(j, r )) J(W(j), r ). ( It follows that NW(z )NFNz N for all. Now since Ve j) z V n! NzN 4! n2j, n4j (n2j)! b(n) the norm V increases with NzN. Thus for all, Ve W(z ( j) ) V/ VF1 and indeed VC W V e F1. This implies that C W is not compact and so the proof is complete. r 41 and! n41 COROLLARY 4. Let p integer j. If C W is compact on H p (b), then NW ( j11) (j)nd1 for all j in U such that W ( j11) (j) exists. n j 41 for some non-negative b(n) R E F E R E N C E S [1] L. AHLFORS, Conformal Invariants, McGraw-Hill, New Yor, [2] J. B. CONWAY, A Course in Functional Analysis, Springer-Verlag, New Yor, [3] W. RUDIN, Function Theory in the Unit Ball of C n, Grundlehren der Mathematischen Wissenschaften, 241, Springer-Verlag, Berlin, [4] K. SEDDIGHI - K. HEDAYATIYAN - B. YOUSEFI, Operators acting on certain Banach spaces of analytic functions, Internat. J. Math. & Math. Sci., 18, No. 1 (1995), [5] A. L. SHIELDS, Weighted shift operators and analytic function theory, Math. Survey, A.M.S. Providence, 13 (1974), [6] B. YOUSEFI, On the space l p (b), Rendiconti del Circolo Matematico di Palermo Serie II, Tomo XLIX (2000), B. Yousefi: Dept. of Math., College of Science, Shiraz University, Shiraz 71454, Iran yousefihmath.susc.ac.ir S. Jahedi: Dept. of Math., College of Science, Shiraz University, Shiraz 71454, Iran Pervenuta in Redazione il 5 marzo 2002