THY-3515 IIIIIIIIIIIIIIIIIIIIIIIIIIIIII 1293 10684 1038 .... LIBn M 153 IGAN STATE UNIVERSITY. ‘WNSING MICH. 4882t This is to certify that the thesis entitled HOLOMORPHIC SELF-MAPS OF THE UNIT BALL: ITERATION AND COMPOSITION OPERATORS presented by Barbara D. MacC1uer has been accepted towards fulfillment of the requirements for Ph . D. degree in Mathematics QM,“ Date May 3. 1983 0-7 639 MSU LIBRARIES RETURNING MATERIALS: Place in back drop to remove this cheCkoUt from your record. FINES wilI be charged if book is returned after the date stamped beTow. HOLOMORPHIC SELF-MAPS OF THE UNIT BALL: ITERATION AND COMPOSITION OPERATORS By Barbara Diane MacCIuer A DISSERTATION Submitted to Michigan State University in partial fquiTIment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1983 ABSTRACT HOLOMORPHIC SELF-MAPS OF THE UNIT BALL: ITERATION AND COMPOSITION OPERATORS By Barbara Diane MacCluer Let f be a holomorphic map of the unit disc D into itself, which is neither the identity map nor an elliptic automorphism of D. A theorem of A. Denjoy and J. Wolff states that the iterates of f converge, uniformly on compacta, to a point z in '5. This point 2, called the Denjoy-WOlff point for f, will be the fixed point of f if f has an interior fixed point; otherwise 2 will be a point of am. In this paper we consider analogues of the Denjoy-wolff theorem for holomorphic self-maps of the unit ball BN in IN. For a fixed point free map f we show that the iterates of f converge to a point of the boundary of the ball. Part of our argument will yield a useful description of the automorphisms of BN which fix no point of BN. The subsequential limits of iterates of maps with interior fixed points are also described. Secondly we consider several questions related to composition operators on Hardy spaces of the unit ball. If m: B + B is a N N holomorphic map and f is a holomorphic function on B denote the N3 composition f o e by C¢(f). We give examples to show that, in con- trast to the situation when N = l, there are holomorphic maps o Barbara D. MacCluer for which Co is not a bounded operator on Hp(BN), where N > l and p < m. Finally, we study Ccp in the case that Co is a compact operator on some Hardy space Hp(BN). In this situation we show that m fixes a unique point 20 of BN and determine the spectrum of Co to be all possible products of powers of the eigenvalues of the derivative map ¢'(zo) U {0,1}. ACKNOWLEDGMENTS I am especially grateful to my advisor, Professor Joel Shapiro, for his guidance and encouragement in my work on this thesis and thoughout my graduate career. He has been a source of inspiration to me, and I have benefitted enormously from having had the opportunity to work with him. I am indebted as well to many other people at Michigan State. I would particularly like to thank Professors Sheldon Axler and William Sledd for several excellent graduate courses, and also Professor Joseph Adney, who, in his capacity as department chairman, has been most helpful and supportive on numerous occasions. ii TABLE OF CONTENTS Page Introduction ......................... l Chapter I. Iteration of Holomorphic Self-Maps of Bn . . . . 4 1. Maps with no interior fixed points ...... 4 2. Maps which fix an interior point ....... 19 Chapter II. Composition Operators on Hp(Bn) ......... 22 l. Notation and preliminaries .......... 22 2. Examples of composition operators which are not bounded on Hp(Bn) ............ 22 3. Sufficient conditions for boundedness of C ...................... 32 9 Chapter III. Compact Composition Operators .......... 36 l. Existence of a fixed point for the inducing map ..................... 36 2. The spectrum of Cg; ............. 4l Bibliography .......................... 50 INTRODUCTION Let BN denote the open unit ball in IN, in the Euclidean metric. Thus BN = {(z],...,zN) 6 CN with leilz < l}. A holomorphic is a map f: B + B which can be written as N N N f = (fl”"’fN) where each fi is a holomorphic function from BN self-map f of B into C. Chapter I deals with the iteration of a holomorphic self-map of BN. A classical result due to A. Denjoy [6] and J. Wolff [24] states that if f is a holomorphic map of the unit disc 0 into itself which is neither the identity nor an elliptic automorphism of D, then the sequence of iterates of f converges, uniformly on com- pacta, to a point z e D} This point 2, called the Denjoy-Wolff point for f, will be the fixed point of f if f has an interior fixed point; otherwise 2 will be a point of 3D. In Chapter I we extend the Denjoy-Wolff theorem to the unit ball BN; in particular we show that the entire sequence of iterates of a fixed point free tolomorphic:self-map of BN converges, uniformly on compacta, to a point in aBN. Part of our argument involves a characterization of We also describe the automorphisms of B which fix no point of B N N‘ the subsequential lhnitsof iterates of maps with interior fixed points. The remaining chapters deal with composition operators on Hardy spaces in the ball. Recall that the Hardy space HP(BN), 0 < p < w, is the space of holomorphic functions on BN satisfying 11:12 sup I 1mg)!” dam .., l. We give, in contrast with the case N = l, some examples of maps o for which Co is not bounded on HP(BN), p < w, and we see that Co can even fail to take HP(B into the N) Ne anlinna class N(BN). Some positive results related to the question of the boundedness of CW are also discussed. Chapter III deals with composition operators which are compact on some HP(BN). Compact composition operators on the HP spaces of the unit disc have been extensively studied. For example, in [20], J.H. Shapiro and P.D. Taylor relate the compactness of the operator Co to certain geometric properties of the inducing map o. J. Caughran and H. Schwartz [4] show that if Ccp is compact on H2(D), then m has a unique fixed point 20 in D, and that the spectrum of Ccp is the set {m'(zo)n: n = l,2,...} U {0,1}. The main object of Chapter III is to determine the spectrum of a compact composition operator on HP(B for l 5 P < m. While the N) fixed point set of a holomorphic map o: BN + BN 15 usually more complicated in the case N > l than it is when N = l, we nevertheless show that a map which induces a compact composition operator on HP(BN) has a unique fixed point in BN’ Suppose that Co is compact on HP(BN) with z the fixed point of m. Then we show that the 0 spectrum of Co consists of O and l along with all possible products of powers of the eigenvalues of the derivative map ¢'(zo). While the methods of Caughran and Schwartz [4] in the case N = l and P = 2 can be made to work for P = 2 in several dimensions, we will take a different approach which is applicable for all P, l f P < m. CHAPTER I In this chapter we consider the iteration of holomorphic self- maps of the unit ball in IN. The main result of Section l is an analogue of the classical Denjoy-Wolff theorem in the unit disc [6, 24], which states that the sequence of iterates of a holomorphic, fixed point free map of m into m converges, uniformly on compacta, to a con- stant of norm l. In Section 2 we consider the iteration of maps with interior fixed points. 1. Maps with no interior fixed points. From now on we will denote N the unit ball in I by B instead of BN’ unless we wish to indicate the dimension explicitly. Let H(B;B) be the family of all holomorphic maps of B into itself. For f E H(B;B) we denote the iterates of f by fn: f = f, f = f o fn n = 1’2939--° n+l Since H(B;B) is a normal family, every sequence of iterates of f contains a subsequence which converges, uniformly on compact subsets of B. We will examine the possible subsequential limits of {fn} according to the fixed point character of f. Note that a subsequential limit of iterates of f E H(B;B) need not belong to H(B;B). However the following lemma shows that this can only happen if the limit is a constant map of norm 1. Lemma 1.1. Let F: B + B' be holomorphic. Then either F(B) Q B or F(z) a g in 38, for all z in B. Proof: Suppose there is a z in B with F(zo) = g e 38. Set 0 G(z) = (l + )/2, so G belongs to A(B), the algebra of functions holomorphic in B and continuous on BI Note that G(;) = l and |G(z)| < l for all z in B\{c}. Consider the holomorphic function G o F. Since G o F(zO) = l and IG 0 F(z)| f l for all z in B, the maximum modulus theorem implies G o F is identically l. So F(z) s g, for all z in B, as desired. D We will find it convenient to use some facts from the theory of topological semigroups. Under the operation of composition and with the topology of uniform convergence on compact subsets of B, H(B;B) becomes a topological semigroup [Zl]. For f E H(B;B) denote by F(f) the closure, in the space of all holomorphic maps from B to CN with the topology of uniform convergence on compact subsets of B, of the iterates of f. If F(f) 9 H(B;B), then F(f) is a compact topological semigroup and as such contains a unique idempotent [23]. Recall g is an idempotent if g o g = 9. An idempotent in H(B;B) is also called a retraction of B. We next give the statement of a theorem due to W. Rudin, which characterizes the fixed point set of any map in H(B;B) as an affine subset of B. This theorem is the key to the proof of the main theorem of this section. Theorem 1.2 [l7; Sec. 8.2.3, p. l66]. If F: B +-B is holomorphic, then the fixed point set of F is an affine subset of B; that is, N the intersection of B with c + L, where c 6 C and L is a com- plex linear subspace of CN. Denote by Aut B the group of biholomorphic maps (automorphisms) of B onto itself. These maps take affine subsets of B onto affine subsets [l7, Sec. 2.4.2, p. 33]. Moreover, since Aut B acts tran- sitively on 8 [l7, Sec. 2.2.3, p. 31], if A is an affine subset of B there is a w 6 Aut B so that w(A) = {(21’22""’ZN) E B with Zi = O for i = r + l,...,N}. To see this, first map some point of A to the origin, so that the image of A is the intersection of B with a complex linear subspace of C”. Now apply a unitary transformation. Thus w(A) 2 Br’ the unit ball in Cr. We will say f E H(B;B) is an automorphism of A if u o f o u'] is an automorphism when restricted to ¢(A). Before stating the main result of this section, we need to develop a several variable analogue of a theorem which in the disc is due to J. Wolff [25]. To facilitate the statement of this theorem we introduce some notation. Let e1 = (l,O,...,O) = (l,O') 6 38. For A > O, E(61,A) = {z = (21.22,...,2N) so that l1 - z1|2 < A(l-Izl2)}. Some computation shows that E(e1,A) is the set of points (Zl’22""’ZN) = (21,2') in EN satisfying Iq-(i-ofi+chw2<¥ where c = A/(l+x). Thus E(e],x) is an ellipsoid in B, centered at e1/(l+A) and containing e1 in its boundary. For an arbitrary c in BB, E(;,x) is the analogous ellipsoid in B, centered at c/(T+A) and containing c in its boundary. Theorem l.3. If f is in H(B;B) and fixed point free, then there is a unique point c 6 38 such that each ellipsoid E(;,A) is mapped into itself by f and every iterate of f. Proof: Choose rn + I. Let an E B be a fixed point of the map rnf: rnB'+ rnBl Passing to a subsequence if necessary, assume an + g e B} Since f has no fixed points in B, g e 38. Without loss of generality assume c = e]. Then an + e], f(an) = an/rn + e], and l-|f(an)| _ l - (Ianl/rn) — < l - lanl l - lanl Again passing to a subsequence if necessary we have l - |f(an)| lim =afI. n+w l - |an| By Julia's lemma [17; Sec. 8.5.3, p. l75] ll-f](z)I2 Il-z]|2 < 0. 14112112 ’ 1-1212 (here f = (f1,f2,...,fN)). Geometrically this means f(E(e1,A)) 9 E(e],aA) 9 E(e],x) since a f l, as desired. To see that c is unique suppose we have another point c' in 38 with the property that each ellipsoid E(c',A) is mapped into itself by f. Choose A] and A2 so that E(;',A]) and E(;,A2) are tangenttn each other at the point z in 8. Then f(z) is in ET;',A]) O ET;,A2) = {z}, contradicting the hypothesis that f is fixed point free. Notation: We call the point t of Theorem l.3 the Denjoy-Wolff point of f. The constant map 9(2) 2 c for z in B will be denoted C(f). A consequence of Theorem 1.3 is the following: Corollary 1.4- LEt f e H(B;B) be fixed point free. Then F(f) contains at most one constant map, which can only be C(f). Proof: Let g be the Denqu-Wolff point of f. Suppose there is a sequence {"i} so that fn + w e BI If w # c we can find a small 1 neighborhood V of w in B disjoint from some ellipsoid E(;,A). By Theorem 1.3, if z is any point in E(;,x), then the image point fn(z) is in E(C:A) for all n 3 l. Thus fn (z) E V for any i, i so fn (z) i w. Therefore the only constant function which may appear 1 in F(f) is ;(f). D We can new state the main theorem of this section. Theorem l.5. Let f be in H(B;B) and suppose f has no fixed points in B. Then fn + §(f). We give the proof of Theorem l.5 in several steps, beginning with the following proposition. Proposition l.6. Let f be an arbitrary map in H(B;B). If there is a nonconstant map among the subsequential limits of {fn}, then F(f) contains a nonconstant idempotent. Proof: We suppose there is a nonconstant map 9 and a sequence 5; = .. {n1} so that fni + 9. Note that g(B) B. Set mi n1+1 n1. Choose a convergent subsequence of {fm }, say fm. + h. On the one 1 1k hand fm. o fn. + h o 9. But also fm. o fn. = fn. + 9. So 1k 1k 1 1 1+1 h o g = g which implies that h is the identity map on the range of g, which consists of more than one point. By Theorem l.2 the fixed point set of h is an affine subset A of B. The dimension of A is 3 l and the range of h contains A. Now suppose that the range of h properly contains A. Then the above argument, applied to h instead of 9, produces another subsequential limit of {fn} which is the identity on an affine subset TO A' of B containing the range of h. Moreover the dimension of A' is strictly greater than the dimension of A. Choose from among the subsequential limits of {fn} a map H with fixed point set of maximal dimension. For this map H we must have range H = fixed point set of H, since otherwise there would be another subsequential limit with a fixed point set of larger dimension. Thus H is an idempotent, and since the dimension of the fixed point set of H is 3 l, H is nonconstant. B Our next goal is to establish Theorem l.5 for automorphisms of B with no fixed points in B. If f is in Aut B then f is continuous from B' to B' and thus has a fixed point in B} The automorphisms of B with no fixed points in B fix either exactly one or exactly two points of 38 [l7, Sec. 2.4.6, p. 33]. The case of two fixed points in 38 is easy to handle: Proposition l.7. Let f E Aut B fix precisely two points of BB. Then fn converges to one of these fixed points. Pgoof: Suppose that f fixes g] and ;2 in 38. Consider the complex line L through ;] and g2. Since an automorphism takes complex lines to complex lines, f maps L O B onto Lil B. Now the Denjoy-Wolff theorem in one variable implies that the iterates of f restricted to LII B converge to one of the fixed points, say g]. By Lemma 1.1, every convergent subsequence of {fn} must converge to g]. This implies that fn + :1, since H(B;B) is a normal family. Clearly ;1 must be the Denjoy-Wolff point of f. D ll The case of one fixed point in BB requires more work. We will assume, without loss of generality, that the fixed point is e1 = (l,O'). To study automorphisms of the disc it is convenient to transfer to the upper half plane via the biholomorphic map 2 + i(l + z)/(l - z). A similar device is available in several variables. N Let n c: C be the region (The Siegel upper half-space) consisting of those points (w],w') with Im w1 > lw'l2, where w' = (w2,...,wN), |w'|2 = |w2|2 + ... + |wN|2. Define o, the Cayley transform, on (IN\{z1 = 1} by ¢(z) = i(e1 + z)/(1 - 2]). Then o is a biholomorphic map of 8 onto 9 [17; Sec. 2.3.1, p. 31]. Moreover if §'= n U39. where an = {(w1,w') such that Im w1 = |w'|2}, and 5’ U {m} is the one-point compactification of 5, then defining ¢(e]) = w extends o to a homeomorphism of B" onto E’U {m}. The automorphisms of B with fixed point set {e1} correspond to the automorphisms of n with fixed point set {w}. An example of a class of such automorphisms are the Heisenberg translations, defined as follows. For each b = (b],b') in an set hb(w],w') = (w1 + b1 + 2i, w' + b'). The Heisenberg translations form a subgroup of Aut n, and for b # 0 each hb fixes w only [17; Sec. 2.3.3, p. 32]. By a Heisenberg translation of B we shall mean an automorphism of B of the form o'] o hb o o, where o is the Cayley transform and hb is as above. It is easy to see that the iterates of a Heisenberg translation converge to e], since for any 0 f b 6 an. (hb)n + m. However, in contrast to the situation in one variable, not every automorphism of Q with fixed point set precisely {w} is a Heisenberg translation. 12 For example, if A = (A2,...,AN) where Ixil = 1 and if b i O is real then gb’x(w1,w') E (w1 + b,x2w2,...,waN) is an automorphism of n fixing {w} only. Note that gb,x fixes setwise the image under o of the complex line through 0 and e1, namely {(w1,w') e n with w' = 0}. We will see that any automorphism of B with fixed point set {e]} which is not a Heisenberg translation of B must fix setwise some nonempty, proper affine subset of B. A map f is said to fix a set S setwise if f(S) 9 S. In this situation we will also say f fixes S as a set. Theorem 1.8. Let G E Aut B fix e1 only. Write G = (Gl’GZ’°"’GN)' If (*) I1 - 61(21l2/(1-iG(-Z)|2)= Il- Z1lz/(l - IZIZ) holds for every 2 in B, then either G is a Heisenberg translation of B or G fixes as a set a proper, nonempty, affine subset of B. Romogk, Professor David Ullrich has pointed out to me that condition (*) of Theorem 1.8 mo§t_hold for an automorphism of B with fixed point set precisely {e1}. We give a proof of this fact at the end of this section. Note that (*) has a simple geometric meaning: the boundary of each ellipsoid E(e],x) is mapped into itself by G. Before giving a proof of Theorem 1.8 we will establish the following corollary. Corollary 1.9. If G e Aut BN fixes e1 only then Gn + e]. 13 Proof. If condition (*) of Theorem 1.8 fails to hold for some point w in B then by Theorem 1.3 we must have 2 I I1- G](W)|2/(I - lG(w) )= eh - w1I2/(1- lez) for some 8 < 1. Suppose further that Gn does not converge to e1 = ;(G). By Corollary 1.4 and Proposition 1.6 F(G) contains a nonconstant idempotent. Moreover, by a theorem of H. Cartan [15; p. 78] the nonconstant subsequential limits of the iterates of an auto- morphism must again be automorphisms. Since the only idempotent which is an automorphism is the identity map I on B, F(G) contains I. Thus there is a sequence {nil so that Gni + I. In particular Gn (w) + w. But this cannot be, for w lies in the boundary of 1 E(e],A) where A = |1 - w1|2/(1 - lwlz) and Gn(w) is in E(e1,BA) 9 int E(e],x) for every n 3 1. This contradicts our as- sumption that Gn does not converge to e]. We suppose now that G satisfies (*) at every point of B and apply Theorem 1.8. If G is a Heisenberg translation o'] o hb o p, then Gn + e1 since (hb)n + m. We finish the remaining case by induction. Note that the corollary is true for N = l and assume it holds for k < N. We are left to consider the possibility that G fixes setwise a nonempty, proper, affine subset A of B of dimension k < N. Now G = GIA is an automorphism of A e Bk fixing e1 only. By induction G” + e1 and by Lemma 1.1 Gn + e]. D In the proof of Theorem 1.8 we will transfer back and forth between the ball 8 and the Siegel upper half-space 9 via the Cayley 14 transform o. For the proof of Theorem 1.8 we will use lower case letters to denote automorphisms of n and the corresponding capital letters for the associated automorphism of B obtained by composition on the right and left by p and o'] respectively. Proof of Theorem 1.8. Let G E Aut BN fix e1 only and satisfy (*). If ¢(z) = w then Im w1 - |w'|2 = (l - |z|2)/|l - 2112. Thus the boundary of the ellipsoid E(e1,x) is mapped by a to {(w],w') E n such that Im w1 - lw'|2 = l/x}. Condition (*) for G e Aut 8N be- comes, for the function g = o o G o c'], on mgnm-Irwn2=mw1-w .l2 where g = (91’92’°"’9N) = (9],9'). Set G(O) = a so g(i,O') = ¢(o) = (a],a'). Now Im a1 ~ |a'|2 = 1 since 9 satisfies (**). Write a1 = c + i(l + |a’|2) where c is real. We claim that there is a Heisenberg translation of 9 taking (a],a') to (i,O'). To see this consider the point (c + i|a'|2,a') in an. The Heisenberg translation associated to this point takes (i,0') to (a],a'). Its inverse is a Heisenberg translation having the desired property; we denote it simply by hb. (A computation shows that b = (-c + ila'lz, -a')). Now hb o g is an automorphism of Q fixing m and (1,0'). The corresponding automorphism F of B fixes O and e], and is just Hb o G. Note that F is unitary. Moreover, since F fixes e], F fixes as a set the orthogonal complement of the complex line through e1, namely the set {2] = 0}. Thus F 21’22""’ZN) = F(z],z')= 1 ( (2],Uz') where U is a unitary operator on CN' . An easy computation 15 shows that -1 . ”1'1 2 . -1 . F° (WI’W)=W’WUW °F(W-|,W) on CN\{w1 = -i}. Therefore the automorphism f of 9 defined by f = o o F o o'] coincides with the original unitary map F on n; f(w) = (w],Uw') (w = (w],w') e D) . At this point we consider two cases. If every eigenvalue of U is 1 then U, and hence F, is the identity. Thus G = Hg] is a Heisenberg translation of B and we are done. So we suppose that U has an eigenvalue e16 f 1. We will show that this implies that G fixes setwise a proper affine subset of B. It is sufficient to 1 show that g = o o G o o' fixes setwise a proper affine subset of 9, since o preserves affine sets. Choose 0 # A = (x2,x3,...,xN) so that A(U) = eIQA where (U) denotes the matrix of the operator U relative to the standard basis on CN']. Recall that g = o o G o 6’] = o 0 HB1 o F 0 ¢‘1 = hB] o f, where h;] is the Heisenberg translation associated to the point (c + ila'lz, a2,...,aN) in an. Let A be the column vector t _ N . (a2,...,aN) so that AA — 122 Aiai' Now con51der the set N ie I = {(w],w2,...,wN) 6 Q w1th 122 kiwi = AA/(l-e )} . I is a nonempty, proper affine subset of Q. We claim that g fixes I as a set. To see this choose (w1,w2,...,wN) in I. Now 16 -l . - . . . g(w],w2,...,wN) = hb o f(w],w ) = hb1(w],Uw ). Wr1t1ng W' = (w2,w3,...,wN)t we see that the last N-l coordinates of hB](w1,Uw') are ((U)W' + A)t. To check that g(w],w2,...,wN) is in I we compute A((U)W' + A) = e1eAW' + AA = e1eAA/(l - e19) + AA = AA/(l - e19) . Therefore g(w],w') is in 3, as desired. U A final observation before the proof of Theorem 1.5 is the following. Lemma 1.10. If f E H(B;B) is such that fn + I, the identity map 1 on B, for some sequence {ni}, then f e Aut B. Proof: We may assume fn _] + 9. Then fn _] o f + g o f. Since fn + I we have 9 o f = I. In particular 9 is in H(B;B) and 1‘ therefore we also have fn = f o fn _] + f o 9. So f o g = g o f = I i i as desired. D Proof of Theorem 1.5. Proposition 1.7 and Corollary 1.9 together establish Theorem 1.5 for automorphisms of B with no fixed points in B. Now suppose f is an arbitrary fixed point free map in H(B;B). If every subsequential limit of {fn} is constant then by Corollary 1.4 fn + g(f), uniformly on compact subsets of B, and we are done. 17 Hence we suppose there is a nonconstant map among the subsequential limits of {fn}. By Proposition 1.6 there is a sequence {"1} and a nonconstant idempotent h so that fn + h. Let A be the fixed 1 point set of h, an affine set of dimension 3 1. We claim that f maps A into A. To see this choose 20 But in A. Now fn (20) + h(zo) = 20 and thus f(fni(zo)) + f(zo). 1. f(f (20)) = fni(f(zo)) + h(f(zo)). So f(zo) = h(f(zo)); that is, f(z is in the fixed point set A of h, as desired. 01 Moreover, fn restricted to A converges to the identity 1. on A. Lemma 1.10, with A replacing B, implies that f s f|A is an automorphism of A, which clearly has no interior fixed points. By Corollary 1.9, In converges to a constant in 3B. But this con- tradicts the fact that Tn converges to the identity map on A. «i Thus the subsequential limits of {fn} must all be constant and we are done by Corollary 1.4. We finish this section with a proof of the fact that condition (*) of Theorem 1.8 must hold for any G E Aut B with fixed point set {e1}. As previously remarked this is equivalent to the following: Theorem 1.11. Let g E Aut 9 fix w only. Then for every w = (w],w') in n (**1 Im 91m) — |g'(w)|2 = Im w] - 1w'|2. Proof. Suppose g(i,O') = (a],a'). Set t = Im a1 - |a'|2. Since (a],a') is in n, t is positive. For 5 > 0 define as E Aut n by D 18 65(w],w') = (52w],sw'). If s f 1 the fixed point set of as is {0,m}. Consider the automorphism as o 9 where s = l//t. The image -1/2 of (i,0') under this map is (t'1a1,t a') and Im(t'1a]) _ tTIIa'IZ = 1. Thus, as in the proof of Theorem 1.8, there is a Heisenberg translation -1 1 he o 6 o g fixes (i,0') and w. Moreover we must so that hc s have, for some unitary operator U, -1 . _ 1 hC ° 65 ° 9(W],W ) ' (W],UW ) so that g(w ,w') = 6 o h (w ,Uw') 1 1/1',— C 1 (t(w1 + c1 + 2i),/t(Uw' + c')). If t = 1 we have g(w],w') = hc(w1,Uw') and an easy com- putation shows that 9 satisfies (**). Suppose that t f 1. We will show that this contradicts the hypothesis that g fixes w only by producing a point in 30 fixed by g. If t f 1 we may solve /E(Uw' + c') = w', since U - t’l/ZI is nonsingular. Let v' denote the solution. If v.l = a + ilv'l2 where a is real, then (v],v') will be in an. We claim we may choose a so that g(v],v') = (v],v'). By our choice of v' we have MwWw=) = a + ilv'lz. Since (v],v') is in an and g is an automorphism, g(v],v') lies in an. 19 Thus for any real a, . I 2 o | | _ I 2 _ ° l 2 Im t(a + 1|v | + c1 + 21) - |v | — Im(a + 1|v | ). Thus (v],v') will be a fixed point of 9 if a is chosen in T1 to satisfy Re t(o + i|v'|2 + c1 + 2i) = a = Re(a + i|v'|2) or to + Re t(c1 + 2i) = a . Since t f l we may solve this equation for real a. Thus the as- sumption that t # 1 implies that the fixed point set of 9 contains more than one point, contradicting the hypothesis. D 2. Mops which fix an interior point. We consider now the case of f e H(B;B) fixing at least one point of B. Several remarks can be made about the sequence of iterates of f; we collect these comments together in: Theorem 1.12. Let f E H(B;B) have a fixed point in B. Then either (1) There is a constant function 9(2) 5 20.6 B in F(f). In this case fn + g, and the fixed point set of f is of course precisely {20}. or (2) There is a sequence {mi} such that fm converges to i 20 a nonconstant idempotent h. The fixed point set of h is an affine subset of B which may be strictly larger than the fixed point set of f, even if f is not in Aut 8. Moreover, if f is not an automorphism of B then every subsequential limit of {fn} is degenerate in the sense that its range is contained in an affine subset of B of lower dimension than B. Proof: Suppose there is a sequence {"1} such that fn converges 1’ to a constant function 9. Then clearly the fixed point set of f is precisely the range of 9. We claim fn + g, for otherwise there is a sequence {mi} such that fmi + h, where h is not a constant map. Without loss of generality fmi'ni + k e H(B;B). Then f i i o fni + k 0 g and also fmi-ni o fni = fmi + h. But k 0 g is constant and h is not, which is a contradiction. This proves (1). If there is no constant map in F(f), then Proposition 1.6 shows that there is a nonconstant idempotent among the subsequential limits of {fn}. Moreover, the proof of Proposition 1.6 shows that given any nonconstant subsequential limit G there is a subsequential limit H which is the identity map on the range of G. Thus if the affine subset of B of smallest dimension containing the range of G is all of B, then the identity map on B is a subsequential limit of {fn}. This implies that f is an automorphism of B, by Lemma 1.10. D For an example where the fixed point set of the limit function is strictly larger than the fixed point set of f, let 9 be a holomorphic function on the unit disc, with |g| < 1. Define f on 21 82 by f(z].22) = (-z],g(z])zz). Thus f e H(Bngz) and the fixed point set of f is {(0,0)}. Now f2k(z],22) = (21,gk(z])gk(-z1)22) and ka + h, where h(z],22) = (21,0). We remark that case (2) of Theorem 1.12 can only occur if f acts as an automorphism on some affine set in B of dimension 3 1. Remarks on Theorems 1.5 and 1.12. Some similar results have been obtained by Yoshisha Kubota [l3],using different methods. He does not consider the fixed point free maps as a separate case, and his result does not show that in this situation the entire sequence of iterates converges to a point in BB. Corollary 1.9 has also been independently obtained by David Ullrich. His argument, while similar in spirit to ours, uses the Iwasawa decomposition for g e Aut n as g = w o 6A o hb where hb is a Heisenberg translation, 6A(w],w') = (A2w],Aw') and w is an automorphism of Q fixing (i,0') in 0. He shows that A = 1 if g fixes w only and that ¢(w],w') = (w],Uw') for some unitary N-1 operator U on C The remainder of the argument proceeds as before. CHAPTER II In this chapter we introduce composition operators on the Hardy space HP(BN)- For N > 1, we give some examples of maps which fail to induce bounded composition operators on any HP(BN) for p < m, and discuss some other results related to the question of boundedness. 1. Notation and Preliminaries. For cliaholomorphicmap of BN into BN and f aholomorphic function on B the composition f o W is denoted by C f). In N’ w( the case N = l, Littlewood's subordination principle [8] shows that Ccp is a bounded linear operator, called a composition operator, on the Hardy space HP(I)), for each p > 0. However for N > 1 this need no longer be the case; in fact we will show that there are maps 4: BN + BN so that for each p < w there exist functions f E HP(BN) for which f o W is not even in the Nevanlinna class N(BN). 2. Examples of composition operators which are not bounded on HP(BN). It is convenient at this point to introduce certain spaces of holomorphic functions in the unit disc I), and examine their con— nection with the spaces HP(BN). Definition 2.1. For a > -1, the weighted Bergman space Ap’a(I)) consists of all holomorphic functions f on I) for which 22 23 211 1 . “f“Ppa i i live“) Ipu - r2)“ rdrde < .. . ’ O 0 Functions in certain weighted Bergman spaces arise naturally in the study of Hp functions in the ball in several variables. In particular, we have the following result: Lemma 2.2. [17; Sec. 1.4.4, p. 14]. If f is a holomorphic function in BN which depends only on the variable 2], then, if fe denotes l the slice function on I) defined by fe (A) = f(Ael), 1 11:11Hp = c11re111Ap,N_2 where c is a constant depending only on N and p. We wish to obtain an extension of Lemma 2.2 to arbitrary functions in Hp(BN). To do this we need the following preliminary result: Lemma 2.3. Let F be in Hp(B and define f on BN by 1 N f(z],z') = F(z],O). Then f is in Hp(BN) with Hpr 5 HFHp. Proof. The argument we give is basically that given in [19; p. 247]. For c in aBN and z in Z“, the closed polydisc in CN, define _ P w(z,;) - |F(Z]c],....ZNcN)| . For each ; in aBN, wc(z) E w(z,c) is an N-subharmonic function in AN, that is, wC is subharmonic in each variable separately. 24 Define W(z) = J33 w(z,g)do(;). Since 0 is unitarily in- N variant, W(z],...,zN) = W(|21|,...,|zN|). This, combined with the N-subharmonicity of W, shows that for all r < 1 W(r,O,...,O) f W(r,r,0,...,0) f ... f W(r,r,...,r) . But W(r,O,...,O) = J |F(r;],0,...,)|pdo(;) = 1 [f(rc)|pdo(c) BBN BBN and W(r,r,...,r) = J [F(rc)lpdo(;), so that Hf” 5 ”F“ as aBN Hp Hp desired. D Lemmas 2.2 and 2.3 together yield: Corollary 2.4. If F is in Hp(BN), N 3 2, and Fe is the slice 1 function defined on I) by Fe (A) = F(Ael), then Fe is in Ap’N'2(I)) 1 1 and “F“ > cUF H . Hp ‘ e1 Ap’N'2 We can now give some examples of holomorphic maps o: BN + BN’ where N > 1, which do not induce bounded composition Operators on any Hp(BN), for l f p < m. Our first result concerns maps e de— fined as follows. Let a be any multi-index a = (01,02....,GN), where a. is a non-negative integer and at least two of the ai's 1 are nonzero. Define ¢(z) = (C(o)2a,0,...,0) where a./2 (*) c 1 (at least two of the indices aj are nonzero) then “9n o o“: + m as n + m. Thus C¢ is not bounded on Hp(BN), since gn is in the unit ball of P H (BN). D Remarks on theoproof of Theorem 2.5. In the case p = 2 we 21 may explicitly exhibit functions F in H BN) for which F o o is not in H2(BN). Choose constants Ck satisfying oo (1) I Ick12(k+1)'”*‘ 0. To see this note first that Ap,N-2 C Ap/Z-e, (N'3)/2. (See [10] for containment relations between weighted Bergman spaces). If f is in Hp(BN), we have f o o = f o w, where f is the restriction of f to the complex line [e1] through 0 and e1 = (1,0'). Since f is in Hp(B ), i is in AP’N‘Z, and N-3)/2. N thus also in Ap/2'€’( Ahern's result now shows that f o o = f o w is in Hp/Z'E (B , as desired. N) Recently A.B. Aleksandrov [2] and E. Low [12] have indepen- dently shown the existence of non-constant inner functions on BN, for N > 1. An inner function in B is a function f 6 Hm(B) whose radial limits f* satisfy |f*(c)| = 1 for almost every c e S. The existence of such functions gives a way of constructing maps o: BN + BN (N > 1) for which C¢ is not bounded on Hp(BN), and moreover, C¢(Hp(BN)) $ N(BN), where N(B is the Nevanlinna class N) consisting of all holomorphic functions in BN satisfying sup J log+ lf(rg)|do(§) < w . O 1, and define o to be an inner map of BN into BN by ¢(z) = (u(z), o,...,o). Then for any p < m, C¢(Hp(BN)) ¢ N(BN). .Proof. By a theorem of Bagemihl, Erdos and Seidel [14, Theorem 4], for any p < w, there is a function g E Ap(D) such that Igl assumes arbitrarily large values along any curve in D which tends to 3D. Extend g to a function defined on BN by setting G(21,z') = 9(21)‘ Note that G is in Hp(BN). For almost every t E S, 4(rc) is a curve in [e]] n BN z D tending to a point of an as r i 1. For each such ;, sup IG 0 ¢(r;)| = w. Since a function in the O -1 let Ap’a(B be the N) space of all functions 'f holomorphicin BN satisfying 11113,. CN JEN Ii1p 114212): dv(2) < ., N where v is Lebesgue measure on C == RZN , normalized so that 30 v(BN) = l, and cN is a constant depending on N, whose value will not interest us. When a = 0 we write Ap(B) instead of Ap’a(B). We have the following lemma. Lemma 2.8. Let g be alxflomorphic function in BN. If the slice functions 9;, defined by gC(A) = g(Ac) (A e D, c E S) form a bounded family in Ap’“(p) as ; runs through 5, then 9 is in Ap'“(BN). Proof. Changing to polar coordinates we have 1 ( |g(z)|p (1-1212)“ dv(z) = 2N ( rZN'1(1—h2)“or ( |g(r;)|pdo(g) B S O 1 < 2N (O h(1-r2)aor (S lg(rg)|pdo(;). Using slice integration [17; Sec. 1.4.7, p. 15] we have [ 19(rc)lp do(c) = ( do(c) j" lg1Pdo/2h. S S -n Thus n 1 . ( 191211911-1212)“ dv(z) 5 C(N) J do(c) ( 1 lociholeilph(i-r2)aohao Bn S -n O = C(N) AS do(§) Jm lgc(z)|p(1-Izl2)adv(z)- By hypothesis, “ggup a f K, for all g 6 S, so from the last line we see that g is in Ap’a(BN). D 31 We use this lemma in the proof of the following result. Proposition 2.9. Suppose o is a holomorphic mapping of Bn into B of the form o = (o,0,...,0). Then C takes Hp(BN) into N o Ap’N'2(BN), with N/P HC¢(f)Hp,N-2 3 c(N,p) (1_1¢%$§?11 ) “f“p . Egoof. Let f be in Hp(BN), and let F denote the restriction of f to [e1], the complex line through 0 and e]. Then F is in Ap’N'2(D), with “Fup,N-2 f c(N,p) “fup, where c(N,p) is a constant depending only on N and p. Note that (f o o)C = F 0 oc. Moreover, since o; is a holomorphic map of D into D, and F is in Ap’N'2(D), we see that F o o; is in Ap’N-2(D), with 1 + |¢c(0)| N/p . (1-1¢C(O)1 ’ “Flp.N-2 IA “F o (13;le ,N-2 1 + 1 (011 N/p = (1-1o1311 ) “F“p.N-2 ° (This estimate, in the case p = 2 and N—2 = 0 appears in [3]. A similar argument yields the result in the more general form we need.) Thus we have shown that {(f o ¢)C: g E S} is a bounded family in Ap’N-2(D). Lemma 2.8 now shows that f o o is in Ap’N-2(BN), with 1 1 (011 N/p 1 o N T P n hf Pup,N-2 f C(Nap) (I'TPIP11 1 thp ' 32 3. Sufficient conditions for boundedness of C3; In spite of the examples of the last section, there are still many interesting examples of bounded composition operators on Hp(BN). In particular, if o is an automorphism of BN’ then Ccp is a bounded operator on every Hp(BN) [17; Sec. 5.6, p. 85]. The next proposition gives a necessary and sufficient condition for Co to be a Hilbert- Schmidt operator on H2(BN). In particular 9? will be bounded, and in fact compact. The case N = 1 of this result is in [20, Theorem 3.1]. Proposition 2.10. O? is a Hilbert-Schmidt operator on H2(BN) if and only if m satisfies [5 [1-1¢(c)l]'N do(c) < w . Egoof, The functions ea = c(a)za form an orthonormal basis for H2(BN), where a is a multi-index a = (a],...,aN) of non-negative a Cl integers, 2“ denotes 211"'ZN N and N-17: (I1: . (101:2aj: (1:: a]. ...aNI) . Thus Co is Hilbert-Schmidt on H2(BN) if and only if 2 a 2 00> 0‘ = Co do guea e12 g(SIHcpl °‘|2 do. IIMB [C(a)¢ O [S lgl=n T1 33 a a If e = ($1,...,oN), by ma we mean ¢]]---¢NNo Using the definition of c(a) and the multi-nomial theorem we see that a - ' n lo} [c(a)<1> 12= $17131: (liplz) . =11 Thus we have CD N-l + g 2 m > [5 ”£0 ((N-l):nz: (lwl ) do = ( (1-1e121'” do S Therefore Co is a Hilbert-Schmidt operator on H2(BN) if and only if I (1-1411-N d0 < ”o D S The remaining results of this section deal with situations in which the boundedness of C(P on Hp(BN) for one value of p allows one to conclude that C? is bounded on Hp(BN) for some other values of p. Proposition 2.11. If C is bounded on Hp(BN), then Co is bounded W on an(B for n = 1,2,... . N) Proof. If f is in an(BN), then fn is in Hp(BN). Moreover “f a @1125 = “f” a (pup 5 ”awn“ “Mug, where new“ denotes the norm of Co as a bounded linear operator on Hp(BN). we have ”f o P“ < HC¢H1/n ”funp’ giving the desired result. D . n p = f np S1nce “f “p H “np’ np - 34 P P Proposition 2.12. Suppose Ccp is bounded on H 1(8) and H 2(B), where 1< p1 < p2 < w. Then Cg: is bounded on Hq(B), for all p1 < q < p2. Proof. Denote the Cauchy transform of a function f E L](o) by C[f]. For 1 < p < w, the map T: f + C[f]* is a bounded linear projection of Lp(o) onto Hp(S) : Hp(B) [17; Sec. 6.3.1, p. 991. The hypothesis on Ccp implies that A o Co o A'] o T is a bounded linear map of ij(o) into Hpi(S) : Lpi(S) (i = 1,2), where A denotes the linear isometry of Hp(B) and Hp(S) given by A(f) = f*, the radial limit function. By the M. Riesz convexity theorem [22, p. 179], A o Co 0 A"1 o T is a bounded linear operator on Lq(o), p1 < q < p2. Suppose f is in Hq(B). Then f* E Hq(S), and since T is onto, f* = 1(9) for some 9 E Lq(o). Thus f o m = Co 0 A.1 o T(g). Since EC¢A1(9)]* is in Lq(o), and C¢AT(g) is holomorphic,f'o p is in Hq(B). The closed graph theorem now shows that Co is bounded on Hq(B), as desired. Propositions 2.11 and 2.12 together yield: Corollary 2.13. If C? is bounded on Hp(B) for some p > 1, then Co is bounded on Hq(B) for all q 3 p. In one variable a standard technique for extending results from one Hardy space to another is to use a factorization theorem. While this technique is generally not available in several variables (for example, the set of functions in H](BN) which can be factored as a 35 product of two functions in H2(BN) is a set of first category in H](BN) [9]), the following substitute for factorization, due to Coifman, Rochberg and Weiss, is sometimes useful. Theorem 2.14. [5, Sec. 3] If f is in H](BN)’ then there exist functions 9i and hi in H2(BN) such that f = X g.h i=1 1 l and .2] 119,112 1111.112 5 c 11111 1: for some constant c depending only on the dimension N. With this result we can prove the following proposition. Proposition 2.15. Suppose Co is bounded linear operator on H2(BN)‘ Then CCP is a bounded operator on H1(BN). Proof. Let f be in H](BN). Write f = I gihi as in Theorem 2.14. “"" i=1 (9- Then f o m = 1 11'M 8 ° ¢)(hi 0 m) and we have “f o 9“] f 1 co .2 119,- oeu, 11h, 0e11, : and)? 11111,. n We remark that Propositions 2.11, 2.12 and 2.15 do not completely answer the question of whether one can conclude that Ccp is bounded for oll_ p < w whenever Co is bounded for ooo_value of p < m. In particular, we do not know in general whether Hp-boundedness implies Hp/Z-boundedness, except in the case p = 2 (Proposition 2.15). CHAPTER III In this chapter we consider composition operators which are compact on some Hp(BN). We show that if m induces a compact operator Cw, then o has a unique fixed point 20 in BN. More- over we show that the spectrum of the compact operator Ccp can then be described as the set consisting of all products of powers of the eigenvalues of the derivative map o'(20) U {0,1}. 1. Existence of a fixed point for the inducing map. Our goal in this section is to Show that a map o inducing a compact operator C? on some space Hp(BN) has a unique fixed point in BN' The motivation for the argument we give is a result due to J. Shapiro and P. Taylor [20] which shows that atnlomorphic map of the disc D into itself with an angular derivative at some point of 3D does not induce a compact composition operator. We begin with the following lemma. ApaN’Z Lemma 3.1. Let N be an integer 3 2. Then for f in we have 1 r [0 J-r |f(y)|p(l — r)N‘2dydr 5 h Hf“: N-2 . Proof. Recall the Fejer-Riesz inequality for a function g in Hp(D) [8; p. 46]: 36 37 1 1 1g1pdx 5 1711911" -1 Hp Now suppose that f is in Ap’N-2(D). For r < l the function fr(z) E f(rz) is in Hp(D). Thus the Fejer-Riesz in- equality gives ] P P (3] |fr(x)| dx 5 n Ufrqu . )N-2 Multiplying this inequality by r(l - r and integrating with respect to r yields the desired result: 1 r 11-2 (0 J_r |f(y)|p(l - r) dydr 5 n “ruip,N_2 . m Now suppose that o is a holomorphic map of B into B with no fixed points in 8. Then o has a (unique) Denjoy-Wolff point t in 38. Recall that this is the point to which the iterates of o converge [see Chapter I, Section 1]. Without loss of generality assume that the Denjoy-Wolff point is the point e, = (l,O'). In this case we have (by Theorem 1.3): lim inf (1 - |o(z)|2 1/(1 - 1212) = a 5 1 2+6] and 11 - ¢1(z)|2 11 - 21 2 < 1 - I¢(Z)| 1 - Izl where P = (91.92,...WN). 38 The next lemma shows that (l - ¢1(re]))/(l - r) is bounded for -1 < r < 1. A similar result appears in [17; Sec. 8.5.6, p. 177]. Lemma 3.2. Let p: B + B be a holomorphic, fixed point free map with Denjoy-Wolff point e]. Then there is an M < m so that l -¢](re1) 1 - r I f M for -1 < r < 1. 2 Proof. Let sup 11 - o](z)| (1 - 1212)/11 zeB z]|2(1 - I¢(2)Iz) = A By the preceeding paragraph we have A g a 5 1. Note that 2 (1) 11 - ¢](re])l 5 A11 - |r(re])lz)(1 - r12/11 - r2) . Thus 2 1 ‘ 181(“911l . 1 + r < 11 ‘ 1’1“"‘911l . 1 - r2 - - 2 2 1 r 1 + ]¢](re]1l 1 _ 1W](re])| (1 _ r) 1 - | 1. The proof uses a several variable analogue of a criterion due to H. Schwartz for a composition operator to be compact. Proposition 3.6. [18; Theorem 2.5]. cm is compact on Hp(B) (l 5 p < m) if and only if for every sequence {fn} bounded in Hp(B) with fn + f uniformly on compact subsets of B, then fn o o + f o o in Hp(B). Proof. The result follows, exactly as in the one variable case, from the fact that {fn o o} is a normal family if {fn} is a bounded sequence in Hp(B). To see this, use the estimate [17; Sec. 7.2.5, p. 128] If, . e i, and 1 Di¢i(0) = aii‘ Step 1. Recall that the nonzero points in the spectrum of a compact operator are always eigenvalues. We will first show that if (*1 f o m = A1 for some holomorphic function f, when A r 0,1, is pot_a product of powers of the eigenvalues aii of ¢'(O), then f s 0. Suppose f satisfies (*), and write f in its homogeneous expansion: f(z) = 5 "MS 0 Fs(z); FS(Z) = |o1=s Caz“ where if is the multi-index (j1""’jN)’ then J J N z“ = 21 1 ... 2N N and lo] = I ji. We will show by induction that i=1 FS 0 for every 5 = 0,1,... . Note that evaluation of both sides of (*) at 0 gives 45 and, since A f l by hypothesis, we have f(0) = F0 = 0. Suppose now that FS 2 O for s < n. Thus co s=n+l Since c(z) = o'(0)z + 0(|z| 2) in a neighborhood of 0, equation (*) yields a (**) |a1=n Ca(Az)a = A|a1=n C 20 where A is the matrix of o'(0) as above. Using the hypotheses on A we will show inductively that Ca = O for every multi-index a with [cl = h. To do this we begin by describing an ordering on the multi-indices with total order n. Suppose a = (j19j290009jN)9 B = (k19k2900-9kN) where ji’ k are nonnegative integers and )ji = in = n. We say i a < B if there is a positive integer i0 such that ji = ki for i < i0 and j, > k, . In particular the first multi-index of total 0 0 order n in this ordering is (n,O,...,O). Comparison of the co- efficients of z“, where a = (n,O,...,O), on both sides of (**) yields n _ c(n,0,...,0)a11 " AC(n,O,...,O) By hypotheses A f a?], so that C(n,0,...,0) = O. 46 Now suppose Ca = O for all a < 8, where [a] = l8] = n. We will show that C8 = 0 by comparing the coefficients of 28 on both sides of (**). Let 8 = (j],j2,...,jN). Since Ca is assumed to be 0 for a < 8 the left hand side of (**) is just (:81th +zcn1Az)" (11 > o. Inl = n) For no n with n > B and In) = n does (Az)n contain a term in 8 z . This follows from the definition of "<" on multi-indices and the fact that A is upper triangular. Thus comparing the coefficients 8 on both sides of (**) we obtain of 2 J- J J 1 2 N _ CBa-l] 322 o o o aNN - AC8. Th1s 1mpl1es C8 = 0, S1nce A f aH a22 ... aNN . Thus we see that Ca = O for all a with lol = n, and hence Fn(z) s 0. The induction on 5 now shows that Fs(z) _ 0 for all s and therefore f 2 O, as desired. Step 2. We complete the proof of Theorem 3.4 by showing that 0,1 and all products of powers of the eigenvalues of o'(0) are in the spectrum of C¢° Since Co is compact, O E 0(C¢) and since f a l is in Hp(B), l is in 0(C¢)' Next we show that aii = Dioi(0) is in 0(Cm)’ for i = 1,...,N. The argument is by induction on i. To see that an is in 0(CQ), we may assume that an f 0. We claim that the function g(z) = 21 is not in the range of (Cm -a]]). For suppose that 47 f o o - a11f = 21 has a solution f e Hp(B). Differentiation gives D f(O)D]o](0) + 02f(O)D 1 1cp2(0) + ... + DNf(O)D]mN(O) - a11D]f(O) = 1 Since Dlo£(0) = 0 for z > 1 this becomes D]f(O)D]m](O) - a1101f(0) = 1 which 15 impossible since 01¢](0) = a1]. Thus aH E °(C¢)' Suppose that ajj e 0(C9) for j = l,...,i-l. We will show that aii is in (CW) by showing that g(z) = 2i is not in the range of (Co - aii)' Without loss of generality we may assume aii f ajj for any j, 1 5 j < 1. Apply the differential monomials Dl’DZ""’Di to the equation f o o Taiif = z, and evaluate both sides of the resulting equation at O. This yields the following: (1) D]f(O)[D]¢](O) -aii] = O (2) D]f(0)DZo](O) + D2f(0)[DZo2(O) -ai,1 = o (i) D]f(O)Dio](0) + ... + 01-1f(O)Dioi_](O) + Dif(0)[Dioi(O) -ai,1 = 1 By the assumption that aii f ajj for any j, l 5 j < i equation (1) implies D]f(0) = 0. Substituting this in (2) shows 02f(0) = 0. Continuing in this manner equation (1) becomes Dif(0)[Di¢i(0) “aiij = l 48 which is a contradiction. Thus every aii’ l f i f N, is in the spectrum of Cw. Moreover, since Co is compact, if aii # 0, then aii 15 an eigenvalue of C? To finish we show that all possible products of the aii's are in C(C¢)' Suppose that A1,...,Am are a collection of aii s, with repeats allowed, and assume that no A1 = 0. We wish to show that TIA- ] is in 0(C¢)' By Lemma 3.7, Co is compact on Hmp(B) and the above argument shows that A1 6 0(Cw)’ relative to Hmp(B). So there is an 0 2 f1 6 Hmp(B) satisfying Thus HA1 6 o(C¢) as desired. This completes the proof of Theorem 3.4. D Remark. Suppose Cm is a power compact operator, i.e., C2 is compact on some Hp(B), for some positive integer M. Since CM = C , where TM denotes the Mth ‘PM P iterate of m, oM must f1x exactly one point of 8. Suppose the fixed point of TM is 20. We claim that o fixes 20. If not, then m fixes no point of 8, since 20 is the only fixed point of ¢M' But then the entire sequence {on} of iterates of o converges to a point ; in BB (Theorem 1.5). 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