'VN 'rui"§" L. .n_;.¢,..,;.uw.. Hyman; . ._.n_..3$.r§ », . a mic... . _.._,,.:.f .... :. wcfibkruanirfir w>rndn2cctrvw Inf" vaulfiauiahaauxtr ‘ - . a. M. 27. THESIS CHSIGAN ST IJIILIHI llzllLll NINIUIUHU)IHNIHIHHIHIIIHllll 301570 7015 This is to certify that the dissertation entitled Composition Operators on The Dirichlet Space presented by William M. Higdon has been accepted towards fulfillment of the requirements for Ph.D. degree in Mathematics WWW Major professor Date 6/23/5 7 MS U i: an Affirmative Action/Equal Opportunity Institution LIBRARY Michigan State University PLACE ll RETURN BOX to remove thie checkout from your record. TO AVOID FINES return on or before dete due. DATE DUE DATE DUE DATE DUE l I l MSU IeAn Affirmetive Anion/Equal Opponunity inetitwon ' Walla-9.1 COMPOSITION OPERATORS ON THE DIRICHLET SPACE By William M. Higdon A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1997 ABSTRACT COMPOSITION OPERATORS ON THE DIRICHLET SPACE By William M. Higdon We examine some properties of functions belonging to, and composition operators acting upon, the Dirichlet and Dirichlet-type spaces of analytic functions on D. Every function in one of these spaces has boundary values on all of 8D except perhaps on a set of capacity zero. We Show that when Cg, is Hilbert-Schmidt, cp may have boundary values of unit modulus only on a set of capacity zero (the converse, of course, does not generally hold). This result is an immediate consequence of an appreciably more descriptive integral condition, which shows that | cp(e“) I cannot be “too big, too often” if C“, is Hilbert-Schmidt. The space ’Do denotes the Dirichlet space modulo the constant functions. We determine the spectrum of each composition operator Cg, on D0 which is induced by a linear fractional map cp taking D into itself. The spectrum of the corresponding composition Operator on the Dirichlet space is essentially the same. To my parents iii ACKNOWLEDGMENTS I thank Professor Joel Shapiro for his guidance, thoughtful suggestions, and pa— tience that he has generously extended to me. I also thank the other members of my Thesis Committee: Professors William Brown, Michael Frazier, Jonathan Hall, and William Sledd. Each of the individuals on my committee has greatly encouraged me, and has taught me much about good teaching and good mathematics during the course of my graduate study. iv TABLE OF CONTENTS 1 Introduction 1.1 Preliminary lemma ............................ 2 The Spaces ’DO and 1),, 2.1 Unitary composition operator theorem ................. 2.2 Reproducing kernel lemma for Do .................... 3 The Spectra of Composition Operators on Do Induced by Linear Fractional 'h'ansformations 3.1 Elliptic automorphism theorem ..................... 3.2 Lemma relating IIFIIW and 13‘ for functions F in H 2(11“) ....... 3.3 Parabolic automorphism theorem .................... 3.4 Hyperbolic automorphism theorem ................... 3.5 A Cauchy lemma ............................. 3.6 Holomorphic semigroup lemma ..................... 3.7 Parabolic non-automorphism theorem .................. 3.8 Hyperbolic non-automorphism theorem (no interior fixed point) 3.9 Hilbert space lemma ........................... 3.10 Hyperbolic non-automorphism theorem (two fixed points in D) . . . . 3.11 A well-known theorem concerning compact iterates .......... 4 Hilbert-Schmidt Composition Operators and Capacity 4.1 Hilbert-Schmidt operator theorem .................... 4.2 Capacity equivalence lemma ....................... 4.3 Kahane and Salem’s theorem ...................... 4.4 Lemma (weak capacitary inequality) .................. 4.5 Factorization lemma ........................... 4.6 Radial limit theorem ........................... 4.7 Corollary of the proof of the radial limit theorem ........... 4.8 Lebesgue point theorem ......................... 4.9 Lemma ................................... 4.10 The main theorem ............................ 4.11 Corollary of the main theorem ...................... BIBLIOGRAPHY 8 9 12 14 17 21 22 24 29 33 34 44 47 47 50 51 53 54 56 59 59 60 62 64 66 CHAPTER 1 Introduction Here and henceforth, cp will denote a non-constant analytic function which maps the unit disk D into itself. The induced composition operator C2,, is defined for each f E H (D) by Ccp(f) = f O (/3- Thus C"p is linear and has range in H (D) In this thesis, the primary concern is on those composition operators which are continuous on the Dirichlet space. The Dirichlet space, denoted by D, consists of all f E H (D) for which / |f’|2dA < 00, D where A is the Lebesgue area measure. D is a Hilbert space with inner product defined for f and g in D by < f,g >i mfg-(07 + i/Df’EdA, and the induced norm . 1 , ||f|l%=| M) l2 ,7 [D I f I2 dA. 2 If f is univalent, then In | f’ |2 (M is precisely the area of f (D) In general, In | f’ |2 dA still yields the area of the image of f on D if one takes multiplicities into account. This area interpretation of the D-norm offers a constructive way to view the space. In Lemma 1.1 below, we prove the well known relation: 1 / If’l2dA = if n | M I2, 7r D ":1 where f(n) denotes the nth Taylor coefficient of f. This provides an alternative formula for the norm: W%=H®V+ZnMWV- Contrasting the formulae for H ”D and || “2, one might expect greater regularity of the functions in D than of the functions in the Hardy space H 2. This does turn out to be the situation, and it is reflected in the theorems of Chapter 4. Briefly stated, capacity tends to play the role in D (and the Do, spaces defined below) that Lebesgue measure plays in H 2. The Do space, a 6 (0,1), consists of all f E H1(D) for which En“ | f(n) |2 < 00. 7121 It is normed by llflli». =| W) I2 + Z n“ | f(n) Iz- n=1 The Do, function spaces are “larger” than the Dirichlet space, “smaller” than H 2, and tend to have “intermediate” regularity. 1.1 Lemma. If f e H(D), then 1 , _°° . ngflidA—gnlflnflz. In particular, f is a member of D if and only if either side, and hence each side, of the equation is finite. PROOF. 1 I2 g/leldA 1 0° * —1 2 — " dA ,, [D | an(n)z | (z) 1 1 21r °° A _/ f I an(n)r"_le("‘l)9 [2 rdOdr 7r 0 0 "=1 1 1 21f Go A 71—1 (n—1)0 2 ;/O/O |an(n)r 26 | deT n=1 1 1 °° . 1 —/ 2W2 | nf(n)r""'5 |2 dr 7T 0 n=1 22722 I f(n) I2] r2"_1dr n=1 0 °° . 1 2 2 _ 2g» |f(n)l 2,, Z” l f(n) |2 n=l /// In his pioneering 1968 paper which examined composition operators and inner functions [7], Nordgren determined the spectrum of Cw as an operator on the Hardy space H 2 when p is an automorphism of D. Cowen has proven many elegant spec- tral theorems, mostly for H 2 and larger spaces (see [2]). Here, we are interested in determining the spectrum of the composition operators C2,, on D which are induced by a linear fractional map (,0. In Chapter 2 we define D0 to be D modulo the constant 4 functions, and we show that the (induced) operator C9,, is unitary on Do when (p is an automorphism. This suggests the space Do as a good starting point, and in Chapter 3 we determine the spectrum of the operator C,p : D0->Do when cp is a linear fractional transformation. We conclude that chapter by observing that the operator C',p : D—>D has essentially the same spectrum as the operator 0,, : D0—>Do. The only difference is that the point 1 is not automatically a spectral value in the D0 case—since the constant functions are identified with the zero element of D0. The eigenfunctions of an operator C,,, on H 2 are often rather abundant, however, just as often they fail to lie in the Dirichlet space. It is due to this, mainly, that the proofs given here are distinguished from those of the corresponding H 2 results. Again, spectral results concerning H 2 appear in Cowen’s work [2] (see also MacCluer’s and Cowen’s book [1, Chapter 7]). In particular, we embrace Cowen’s use of a semigroup in case (p is a parabolic non-automorphism (the idea for which he attributes to R. P. Kaufman [2]), as well as his resourceful application of the invariance of the reproducing kernels under C;. In Chapter 4 the main theorem, Theorem 4.10, is a generalization of the following well-known result ( [9, p.32]). If C“, : H 2—)H 2 is compact, then the Lebesgue measure of the set {8“ : |g0(e“)| = 1} is zero. Theorem 4.10 shows that if C}, is Hilbert-Schmidt on D or D0, then the capacity of the set upon which (,0 has unit modulus ({e“ : |<,0(e“)| = 1}) is zero. CHAPTER 2 The Spaces D0 and D,r Let C denote the class of constant functions in D. Let D0 denote the Hilbert space D/C with the norm and inner product that it inherits from D. That is, ||[f]II%. i if!) I f’ |2 M for f e [f] 6 Do, and < m, [91 >1»; i/Df'rdA for f e if] e 00. g e [916 Do. These definitions do not depend on the representatives chosen and are thus well- defined. Let cp be an analytic self-map of the unit disk for which 0,, : D—>D is continuous (or equivalently, by the Closed Graph Theorem, merely well-defined as a mapping). For any representatives f and g of [f] 6 Do, we have for-QOPEC- This shows that the operator 0,, : D0—>D0 induced by C“, : D—iD is well-defined. The following theorem is a simple consequence of how the D0 norm neatly trans- lates composition by (,0 into a change of variables. 2.1 Theorem. If (p : D—)D is an automorphism, then C1,, : D0—)D0 is an isometric isomorphism (i.e. CV, is unitary). PROOF. For any f 6 D0, “(Janna = if!) |(focp)’|2dA ifiprwmnflwmrdma 1 I 2 =-(@flf®ldfld , =§£¢rmrdaa = IIfIIIDo' —1 Moreover, (,0 is also a disk automorphism and (C,,)‘1 = C,p—1. /// Similar reasoning shows, more generally, that Cw : Do—)Do satisfies ||C,p|| 5 fl whenever (p : D—>D is at most n-valent. Let D,r denote the space of equivalence classes of analytic functions, defined on the upper half-plane 11”“, which is analogous to D/C. More precisely, [F] .—'.- {F(z) +c e H(H+): c e C} e D, if 1 MHMi-leTdAI'I+ induce well-defined Operators 0,), : D1,—>D,,. We will see that many of the composition 7 Operators C1,, : Do—->D0 we consider are similar to simpler composition operators on D,. In the sequel, we will only consider composition Operators C,,, on D, where it : II”'-—>I'I+ is a translation or multiplication by a positive scalar. In these cases, it is very easy to see that C, : D,—+D, is a bounded operator. To simplify notation in the sequel, we will identify any member of Do or D, with any (and Often a particular one) Of its representatives. The statement and proof of the following lemma illustrates this usage. 2.2 Lemma. For 11) E D, the functions Kw(z)ilog( 1_)=§%flz" (zED) are reproducing kernels for D0. PROOF. Let it) E D. Then 00 ZnIR,(n)I2 = in — n=1 n21 — Z lwlzn — lO ——1 n _ g1—|w|2’ so Kw 6 D0. Choose the representative f of [f] E Do with f (0) = 0. Then 1). = inflame) = inflmfi) = imam", n21 n=l so130: f(w). /// CHAPTER 3 The Spectra of Composition Operators on Do Induced by Linear Fractional Transformations In this chapter, we shall determine the spectrum of each composition operator C, on Do induced by a linear fractional map (,0 which takes the unit disk into itself. The Remark at the end of this chapter shows that the spectrum of a composition Operator C, : D—->D is essentially the same as that of C, : Do—-)Do. The only difference is that the point 1 is not automatically a spectral value in the D0 case—since the constant functions are identified with the zero element of Do. For determining the spectrum of composition Operators on D, this is an important reduction. We presume that the reader is familiar with the following ideas: e The elliptic maps are those which are similar to a rotation of the disk. e The parabolic maps are those which are similar to a translation in a half plane. e The hyperbolic maps are those which are similar to a positive dilation in a disk or a half plane. We furnish some explicit examples of these mappings below. See [9], for instance, for more detailed information on the fundamental characteristics of the linear fractional transformations. 9 Some Sample Linear Fractional Transformations of D Define ,u by p(2) +— 511—32. [1. is a linear fractional transformation which maps D onto the upper half plane. It has inverse p’1(w) = ”—3. We use the formulae for ,u and w [1‘1 below. e D is an elliptic automorphism, then the Operator C, : D0—>Do has spectrum equal to the closure of the set {D0 is invertible for every 5 E T\E. ‘ Let E E T\E and set d = dist (E, E). Then 6 > 0. It suffices to show that C, — {I is both surjective and bounded from below on Do. Let f 6 D0. Define 9(2) = E: A132 2" (z e D). Then - f (n) < |f(n)| An—g |§(n) |= _ d for each n, and so Lemma 1.1 shows that g 6 Do. For each n E N, ((C, - EI)9)”(71) = Cw(g)i(n) - €901) = ram—tan) = Mung”), = in). Hence (C, — {1)g = f, and along with (3.1), this shows that C, — E] is surjective and bounded from below. /// We regard H 2(11+) as the set of all f E H (IV) for which sup |f(:1:+z'y) |2 dz < oo. 0 0, Hz) = Ego + m) = — f0” f(t)z'te'y‘e”"dt. 13 By Plancherel’s theorem, for each y > 0, /_°° I f’(;z:+iy) I2 d2: = /°° l f(IIz'te-vt I2 dt. Then since f(z) = 0 for (1.6. a: < O, f...|f’I2 dA = [Om/:II'IAHM dzdy = [0” 1: I f(t)z'te"y‘ I2 dtdy = /0°° I fit) l2 2:2 [DOOW‘dz/dt Division by 7r gives the desired result. /// Remark. The relationship 61/1 1 0° A I2__=_ 2t [mm 7, 2,,fotIfItHd, which we have shown to be valid for all f 6 H 2(11+) 0 D,, holds more generally. To each F E D,, there corresponds a function S in L2([O, 00), t 21:?) with 1 oo 2 :_ t2 ' ”File. 2,], tlS()| dt Moreover, this correspondence is surjective as well as isometric. We omit the proof since we do not have a current need for this generalization. The relatively simple proof will be included in an article which is in preparation for publication. 14 3.3 Theorem. If Ip : D->D is a parabolic automorphism, then the spectrum of the operator C, I D0—>D0 IS T. PROOF. By Theorem 2.1, C, : Do—>Do is unitary and so o(C,) _C_ T. It remains only to prove the other inclusion. There is a linear fractional map )2 taking D onto 11+ and a real number a so that cp = [fl 0 7' o p, where T(’lU) i w — a. Note that the Operators CV1 : D0—9D, and C, : D,—+Do are unitary. Moreover, C, =C,,0C,0C,,—1 =C,,0C,0(C;1) and it follows that 0(C,) = 0(C,). Therefore the proof of the theorem will be complete upon establishing that T Q 0(C,). Fix any point 6‘” [E T, where fl 6 R. We will show that (C, —- e‘”) : D,—>D, is not bounded from below. Choose k 6 R so that 27rka = 0 (mod 2%) and 27rk 2 fi/a. For1>c>0, set [a, b] = [—fi/a + 27rk, —fi/a+ 27rk + c] C [0, 00) (although c will be used as an indexing parameter in this proof, we will refrain from subscripting a and b). Define FC 6 H 2(I'I‘L) by Fc(z) = t) e”t dt 1 oo fi/o Xia.bl( 15 izt dt. 1 b — e v27r [a Then we have F, = XI”, Claim: F, E D, and ||F,|[,2T = (C1 c+c2)/(47r) for a constant C1 which does not depend on c. F, E H2(II+) since F, E L2([O, 00)) ([8, p. 372]). Thus application Of Lemma 3.2 shows that . 1 oo . IIFII: = —/ ch(1t)|2W Zn 0 = 5—” fabtdt = (b2 - a2)/(47r) = [(—fi/a + 2M + c)2 — (—fl/a + 27rk)2]/(47r) = (2(—fl/a + 27rk)c + c2)/(47r) = (Cl C “I" 62)/(47i'). For F E H2(II+), and from this we see that ((CT - e”)FCYOi) = (6""‘ - 6”)Fc(t) 16 = (640‘ _ eifl)Xla.bl(t)' As (C, — ei”)F, E H2(II+), Lemma 3.2 implies that 1 °° —iat m I 2 filo l (e — 8 )X[a,b](t) I tdt 1 b . . = 2—/ lam—em |2 tdt 7i' 0 1 b-a . . ____ fi/ I e—za(t+a) _ ezfl I2 (t-i- a) dt 0 1 c . . = g] [e—’a(t—B/a+2”k)—e'5|2 (t+a)dt (3.3) 0 “(0, - 6”)FcII§ = since b — a = c. As 27rka = 0 (mod 27r), the quantity in (3.3) equals 1 C —iat 2 RA II: —II (t+a)dt. (3.4) For small values Of c, the factor in the integrand, | 6"“ — 1 |, satisfies le‘iat—llx t. Hence, there exists 6 > 0 and a constant C2 so that |e'“"——1|2(t+a)§ Cgt2 when t E [0, 6]. This shows that for each c 6 (0,6), the quantity in (3.4) does not exceed 02 03. Therefore, when c < 6, [[(C, _ ew)FCIII2r S C2 03' 17 By this result and the Claim, for c < 6, we have [[(C, - eifl)F,[[3r C? (33 , lchH'fi — (010+Czl/(47FV and the right hand side tends to 0 as c—>0. Thus the ”Operator (C, — e”) : D,——>D, is not bounded from below, and so 6'” E 0(C,). By the freedom with which we chose 6, it follows that T g 0(C,). /// 3.4 Theorem. If C, : D—-+D is a hyperbolic automorphism, then the operator C, : D0—>D0 has spectrum T. PROOF. By Theorem 2.1, C, : D0—>Do is unitary and so 0(C,) g T. It therefore suffices to establish the reverse inclusion. There is a linear fractional map 11 taking D onto 11+ 1 and a positive number A, A 75 1, so that (p = If 0 7' o u, where r(w) i Aw. As in the previous theorem, by similarity, we have 0(C, : D0—>’Do) = 0(C, : D,—>D,). Fix 6‘” E T for any real number 6. We will show that the Operator (C, — ewl) : D,—>D, is not bounded from below. Define 1 . g(t) = 26"“10‘9‘ (t > 0). 18 Claim 1: %g(t/A) = ei”g(t) for all t > 0. For any t > 0, 1 1 . -g(t/A\) = —(t/A>-1e-W°gA<‘/*> A A = 16—w((log.t)—1) t ___ leifle—ifilogxt t = emg(t). For each value Of c with 0 < c < k i min(A,1/A) (which ensures that [c, 1] (1 [Ac, A] = 0), define F, E H2(I'I+) by Fee) = 712—; [omgm x...(t> dt (2 6 IF). Then we have F, = g XI“, Denote by C, the composition operator on H (II+) induced by 7'. Claim 2: C,(F,) e H2(II+). For any y > 0, [_c: I ICTIFc)I(~"3 +11!) I2 d1" = L: I FC(A:L' +2'Ay) |2 dz 1 00 2 X100 [ F,(:r + iAy) I2 dun. (3.5) Since F, E H 2(11+), the quantity in (3.5) is bounded by a constant which does not depend on the value of y (y > 0). This proves Claim 2. 19 Observe that (C,(F,) —e*‘BF,)“(t) = (Fc(/\x))‘(t)-e”1:‘c(t) = lF,(t/A)—6”Fc(t) = —g(t//\)XI...I(t//\) — 6“}QIIIXIAHIII = Xg(I/z\)xI....I(t) - e”g(t)XI...I(t) = e”g(t)xI..,.I(t) - 6”Q(t)XIc.u(t) H)? H)’ by Claim 1. This function certainly vanishes for t E [cA, A] n [c, 1]. Moreover, if A > 1 then it has support given by the union of the disjoint intervals [c, cA] U [1, A]. If A < 1 then it has support given by the union of the disjoint intervals [cA, c] U [A, 1]. Using Claim 2, we that that C,(F,) — ewF, E H2(I'I+), so by Lemma 3.2 1 27r IIIC. — emu: = [0... I (I0. — ROM) |2 wt. (3.6) As | g(t) |= % for t > 0, by the observations above, the quantity in (3.6) equals 1 27r CA1 1 —dt __ [c t I+27r *1 f—dt| =|lnA[/7r. 1 t Hence for all c E (0,k), ||(C, - e”)Fclli = | 111* l /W- However, for each c E (0, k), > “Fall?r = I C(t)|2Idt I 9(t)XI...,(t) |2 tdt S’IH‘S’IH o\.o\. Therefore, for c E (0, k) “(C-r - e”)EIIi _ —2 l 111A I “Fall?r _ 1116 ’ and the right hand side tends to 0 as c——>0. Thus, the operator (C, —- ewI) : D,—)D,r is not bounded from below, and so we Obtain T Q O'(C,, I D0—>D0). The reverse inclusion was established earlier. /// Towards The Case Where (,0 Is A Parabolic Non-automorphism In [2] (see also [1, Chapter 7]) Cowen constructs, rather generally, a holomorphic semigroup of Operators {Ct},EA on H 2 and exhibits a spiral-like set or segment E, for which 0(Ct) C; Et. Following his method, we construct a holomorphic semigroup of Operators on D0 which we utilize in an analogous way in Theorem 3.7. That theorem deals with the case where I0 is a parabolic non-automorphism. The following lemma, which will be used in the construction of the semigroup, follows from a slightly more general version which appears in [4]. 21 3.5 Lemma. Let f E H (11+) and let S C II+ be a compact subset. Then there exists a number M satisfying 3M 0—5 a 1 [inc + a) — m» — %III< + A) -— f(<>> whenever three different numbers C, C + a, and C + 6 lie in S. PROOF. Let C be a closed path in 11“, having index 1 on S, which satisfies dist (C', S) > 0. By Cauchy’s theorem, f(z) - —1- “”0” 2m C T-Z (z E S). Using this representation for each Of the four occurrences of f in the expression we are taking the absolute value of, the expression becomes _1_/ f(T)dT 2m 0 (r — (g +a))(r -— (C +fi))' Observe that the modulus of the integrand above, and hence the integral, is uniformly bounded for all choices Of C, a, and fl satisfying the hypothesis. This proves the lemma. /// Let p : D—>II+ be an analytic, bijective mapping. For 111 E H+, define Ip, : D—>D by 9010(2) = u"1(u(2)+ 10)- Write C, for C,w, the composition Operator on Do induced by 90,. 22 3.6 Lemma. {Cw},en+ is a holomorphic semigroup of operators on Do. This means that : (a) 01010102 : w1+w2 (1111, 102 E 11+)- (b) w I——> C, is a continuous map into the space of operators on D0 (11) E 11+). (c) For any A E B(D0, Do)“, the function 10 +—> A(Cw) lies in H(II+). PROOF. It is trivial to verify that (a) holds, so we prove (b) and (c). Claim: For each f and g in D0, < C, f, g > is an analytic function Of w, (w E H+). Let f and g be given. Denoting the reproducing kernel at the point p # 0 by K, (see Lemma 2.2), we see that = [wal(p) = f(«pw(p)) = f(u“1(u(p) + w))- Therefore < C, f, K, > is an analytic function of 10, since f and 0‘1 are analytic functions. As the linear span Of the set {K p : p E D\{0} } is dense in D0, there exists a sequence {9, 3:1 in this linear span with gn—>g in D0. The Observations above then imply that < C, f, g, > is analytic in w E 11+, for each n E N. As gn—>g, there exists 23 a constant M, with M _>_ ||g,,||1;0 for all n. Then by the Cauchy-Schwartz inequality, I < waagn > I S IIwaIIDo IIgnIIDo MIIfIIDo . l/\ and so {< C, f, g, >};’,°:1 is a normal family. One easily shows that it has < C, f, g > as a limit point, in the topology of uniform convergence, proving the Claim. Fix C E II+, and choose 7‘ so that B(C,r) C IV. For every 0 and 6 in B(0,r)\{0} with a 75 6, define the Operator U (a, ,6) : D0—+Do by _l I3 1 UIOII fl) = 3(00... - Cc) a—fl (C(+I3_CC) ' Let f,g E Do. By the Claim, h(w) i < C,f,g > E H(II+). Thus by Lemma 3.5, there exists M such that for any a and 6 as above 1'[§wc+m—w«»—% > M 0‘5 (Mc+m—h«flI (an As h(w) =< C,f,g >, (3.7) may be written M 2 l < U(a,fi)f,g > I . By applying the Uniform Boundedness Principle, twice, there exists a constant M2 satisfying IIU(aiIB)II S M2 24 for all a and 5 as above. Equivalently, 1 1 “3(ch — Cc) flIch — C(III S M2 I a — 5 I - (3-8) Define I‘('y) = £(CC+, — CC) for '7 E B(0,r)\{0}. Inequality (3.8) shows that I‘(7) is uniformly Cauchy, in the operator norm, as 7+0 in C. Therefore the following limit exists: , . . . 1 CC 2 11m I‘(7) = lIm—(CC+,—CC). 7—+0 'y—+0 ”y This shows (b), that the mapping w I—> C, is continuous at C (hence on II+). This also implies (c), for let A E B (D0,D0)*. Then lim A(CC+h) — A(C,) = limA (CHI: _ CC) h—>0 h, h—>0 h = A (lim ————C<+" - CC ) h—>0 h = A(Cé) and so w I—> A(C,) is analytic at w = C, hence on II+. ' /// Following Cowen’s work on H 2, we prove the following theorem. 3.7 Theorem. Let <0 : D—)D be a parabolic non-automorphism. Then the operator C, : D0—>D0 has spectrum E: {ei”°t: tE [0,00)} 0 {0} for some number wo E 11+. 25 PROOF. There is a linear fractional map It taking D onto 11+ and a non-real number w0=x0+iyo EII+ so that mzp'lorop, where 7(a)) éw+wo. Claim: 0(C,) C E. For m E 11+, define <,0, : D—+D by PwIz) = #"1(u(z) + w) (z E D)- Write C, for C,w, the composition operator on Do induced by D, and C,, : D,—>D0 are unitary, and since C, = C,,OC,oC,,—1 =C,,oC,0C;1, it follows that 0(C, : D0—+D0) = 0(C, : D,—-)D,). It therefore suffices to show that E Q a(C,). Let A = ei”°‘° E E, for any to E [0, 00). For c > 0, define F, E H2(H+) by F,(z) (t) em dt (2: E 11+). 1 oo : —\/—2'__7;Loo XI‘O-‘O‘I'CI Then we have F, = Xhmmm. By Plancherel’s theorem, (C,F,)A(s) = #1: F,(:r+w0)e““ d2: 1 [00 1 [00 F (t) i(:r:+wo)t dt —isa: d — — , e e a: V27i' -oo V27i' —oo 1 0° 1 00 A iwot imt —isa: “.07; I... 72—, I... We 6 d“ d‘” = F,(s)ei”°8 (m — (i.e.). 28 Therefore, (C,F, — AF,)A(3) = (ei”°s — ei”°t°)F,(s) (eiwos - eiwoto )X1:o.to+c1 (3) As 0,15; —- AF, 6 H2(II+), by Lemma 3.2, 1 00 new. — ARI: = 5,; [0 Hour. — mm) |2 tdt 1 0° iw iwo Z .2—7;/0 I (e at _ e to)X[¢o-‘o+¢l(t) I2 tdt t +c . . = .21? f 0 | (e‘w0t — em") |2 tdt. (3.14) to Define k(t) = e‘”°‘ for t E [to, to + c]. Then the quantity in (3.14) becomes 1 to+c / I k(t) — k(t0) I2 tdt. (3.15) 27f to Note that k(t) —— k(t0) z k’ (t0)(t — to) when t is near to. Therefore, there exists a constant K and 6 > 0 so that IkItl—kItol IAS K2 I 75—I0 I whenever | t — to [< 6. Then for c < 6, the quantity in (3.15) is at most K2 to+c K2 C — t—tztdt=—/t2t tdt 27r ./to ( 0) 27r o (+ 0) K2 c < —t tzdt _ 27r(0+c)/0 S 0163 29 for a constant C1 independent of c (0 < c < 6). This shows that ||C,F, — AF,||,2, g C1c3 whenever c < 6. On the other hand, 1 oo . ”an: = 27/0 ch(t)|2tdt 1 to-i-ctdt - 27/. 1 = EIIto + CIA — t6) 1 = EIAC to + c2) = C2 c + c2/(47r) for all c > 0. Therefore for all c E (0,6), ||C,F,—AF,||,2,< C1c3 . “Fall?r ‘Czc+cz/(47r)’ and the right hand side tends to 0 as c——>0. Thus C, — AI : D,——>D, is not bounded from below. Therefore A E o(C,), and hence {ei”°‘ : t E [0, oo) } Q o(C,). Since the spectrum is closed, this implies that E Q 0(C,). This completes the proof Of the theorem. /// 3.8 Theorem. If (0 : D—>D is hyperbolic with precisely one fixed point on 8D and no interior fixed point, then 0(C, : Do—IDO) = D. 3O PROOF. There exists a linear fractional map )1 and a positive number A, with A 75 1, so that (p = ”"1070”, where T(Z) i Az. IfA > 1, define p(z) = FIT); then p‘1(w) = ,u‘1(1/w), and Therefore, we may further assume that'A E (0, 1). Claim: ”(D) is a circle with the point 0 on its boundary. As it is a linear fractional map, ,u(D) is either a half-plane or a circle. Suppose first that ”(D) is a half-plane. Then there exists a point c E 6D at which [1. is singular. Then (0(0) = p_10TO/I(C) = c and so c is the boundary fixed point of (0. If 0 E ,u(D), then p“1(0) is another fixed point of (,0 (contrary to our hypothesis). Thus 0 ¢ ( ). But then T 0 H(D) = AMD) Z #( ), and this implies that (,0 is not a self map of the disk. Therefore u(D) must be a circle. Reasoning as above, if 0 E ”(D) we Obtain an interior fixed point for (,0; if 0 ¢ p(D), then (,0 is not a self-map of the disk. Each Of these conclusions is con- trary to the hypothesis, and so 0 E 6(u(D)), completing the proof of the Claim. Set P = ,u(D), and denote by D): the space of functions analytic on P which is 31 analogous to D,, i.e. 1 DP 2 {[F]: F 6 mp), Will?» A g [P I F' l2 M < oo}. It is easy to see that C,,—1 : DO—Dp and C,, : Dp—>D0 are unitary Operators. More- over, since C,=C,0C,0C,,—1=C,,0C,0C;l, C, : Do—Do and C, : Dp——)Dp share the same eigenvalues. As (0 is univalent, ”C,“ S 1 and so 0(C, : D0—>D0) Q D. We will show that each 6 E D\{0} is an eigenvalue of C,. Fix 6 E D\{O}. Define the function F, on P by n6 nA F3(z) : zI—‘. Writing 6 = [6 [601, we have FBIZ) = ,(IanRé) lnz)(——MI6I+io ) = 8( 10A Since the logarithm is analytic on P, so is F5. Setting c1 2 [%§, we have F,(Z) : 61201—1: _C_1_ec1lnz fl Z For each 2 2 re” E P, |F[’,(re”)| : |:_lec1(lnr+i0)| I C_le(1n IBI+101 )(ln r+i0)/ in Al r 32 [21' leunwlxlane—ololfi , < E [6(ln|B|)(lnr)|I-,‘: ,. for some constant C2 which does not depend on 2 (z E P). Thus . C n )9 [Fé(re”)| _<_ —A:rl—IFIT1 = Cg'r" r where a > —1, since 1,93% > 0. Choose R large enough so that P C B (0, R). We have F, 2dA < C2] 20 dA [I flI — 2 IZI (Z) < 02/ 2061/4 — 2 3(0, )IZI (Z) R C22 27r / r2°+1dr. 0 |/\ Since 2a + 2 = 2(a + 1) > 0, the latter integral is finite and so F, E Dp. For each zEP, C(ngz) = (MI? CD I lag ln ln A Z In =A V _ e(ln A)(ln Ifi|+i91)/(ln A)Ffi(z) = W BIA‘FMZ) = WW2)- Thus C,(Fg) = ,BFp, and so 6 is an eigenvalue of C,. Because of the freedom with which we chose 3, every point in D\{0} is an eigenvalue. We observed that 0(C,) = 0(C,) Q D. Since the spectrum is a closed set, we conclude that 0(CIp) = 5- /// 33 For the convenience Of the reader, we state the following lemma, which can be found for instance in [1, p. 270]. It will be helpful in the proof of Theorem 3.10, and again at the end Of this chapter. 3.9 Hilbert Space Lemma Suppose H is a Hilbert space with H = K EB L, where K is finite dimensional, and C is a bounded operator on H that leaves K or L invariant. If C has the matrix representation C: or C: We define L1 to be Do. The reproducing kernels for Lm, denoted K,,m, are defined for each 21) E D by z" (z E D). °° III" K,,m(z) = Z — "=77! n Suppose that C, is a bounded composition Operator on Do and that (0(0) :2 0. Then the restriction of C, to L, has its range contained in Lm. Let 0;, denote the adjoint of the Operator C, on Lm. A routine argument shows that the family of reproducing 34 kernels (KW, : w E D} is invariant under C; and that, in particular, 0;,(K,.,) = K,(,),,,,. (3.16) L, has finite codimension in D,, so application Of Lemma 3.9 ensures that 0(C, : Lm—2Lm) Q 0(C, : D0—)Do). (3.17) Following Cowen’s proof of Theorem 7.30 in [1, p. 289], wherein he makes effective use of the H 2 analogues of (3.16) and (3.17), we are able to prove the following theorem. 3.10 Theorem. If (p : D—>D is a hyperbolic map with an interior fixed point (necessarily attractive) and a boundary fixed point, then the spectrum of the operator C, : D0—+D0 is D. PROOF. WLOG, we may assume that (p fixes the points 0 and 1. Hence, by our hypothesis 0 < (0’(0) <1 < (p’(1). Throughout the proof, m and J will always denote positive integers. In accordance with the remarks preceding the theorem, let 0;, denote the adjoint of the opera- tor C, : Lm—Lm. Fix A E D\{0}. To see that A is contained in 0(C, : D0—>D0), by (3.17), it is sufficient to show for some value of m, that A is contained in 0(C, : Lm—>Lm). This is our underlying goal in the remainder. Since (,0 is a linear fractional transformation which fixes 0 and 1, it follows that (0 is 35 a homeomorphism of the interval [0, 1]. For any point a: E (O, 1), consider the sequence {:r,,}$,°=_oo consisting of the forward and backward (p-iterates of 2:, i.e. {2,}$,°=_°o is the uniquely determined such sequence having 51:0 A; :12, whose elements satisfy the family of relations .12..“ = (0(a) (n e Z). (3.18) Let us pause to outline the rest Of the proof. A primary tool in our argument is, from (3.16), that Cg, is a forward shift of the sequence {K,mm}$f= It is not —W' difficult to check, formally, that 00 Z X‘" K,,,,, 712-“) is an eigenfunction of C; corresponding to A. We shall see that for m sufficiently large, this is a convergent series. It is necessary, however, that the series not be zero— if it is to be an eigenfunction. We show in Claim 2, non-trivially, that a sequence of partial sums of the series is bounded away from zero. This lower bound, of course, also applies to the limit. In this way we Obtain an eigenfunction for C; corresponding to A, implying that A E 0(C, : Lm—)Lm). The homeomorphism Of the interval [0, 1] described above, along with the Schwarz Lemma, provides that O<$n+1<$n<1 (REZ). Since (0 has no fixed points in (0,1), this implies that lim 2-, z 1. 71—)00 36 Indeed, since 1_ :17." (10(1) “ 90(33—11—1) 1° —— : 1 = ’ 1 713,20 1 — $_n_1 ”Ego 1 _ $_n_1 ”0 ( ) and (,0’(1) > 1, we have 2(1 —:1:_,,) < oo. (3.19) :0 This shows that the backward iterates of 2:0 tend to 1 quickly enough to be the zeros of a Blaschke product. Let s be a number satisfying 0 < s < 5%. Then 3 < 1, and there exists a number a in the interval (.5, 1) such that 1—2: — Z 3 whenever 1 > a: 2 a. 3.20 1 - 90(27) ( I We now fix the sequence {23,}::_°o determined by $0 = a and the relations given in (3.18). For any value of J, the backward (p-iterates 22-1, 2-2, . . ., :r_J lie in (01,1), and so by (3.20) 1—a:_J 1—:r_J+1 1—a:_1- 1 ‘— _ 2 > 1 — _ = . . . 1 _ (A: J) :1: J 1— $-J+1 1— $-J+2 1— $0 ( $0) _ 1—$-J 1—$_J+1 l—IE._1 (1 :L‘) 1 — (obs—J) 1 — ALA.) 1 — (om—1) ° 2 SJ(1 — 1130). This inequality provides the entire means for the following claim. Claim 1: There are constants M and J0 so that ||K,_J||p0 3 MW whenever J 2 J0. 37 Since “K H2 — Io —— x—J Do — g1 _ ($-J)2 < ___ _ log sJ(1— $0) = Jllogsl —- log(1 — 3:0), we have IIK,_,II,, g \/J|logs| — log(1 — 230). (3.21) Claim 1 follows from (3.21). A simple application of the Schwarz Lemma yields a constant c in (0,1) which satisfies the condition: [90(2)] _<_ c|z| when ([2] g .5). (3.22) Set N = min{n: 11:, g .5}. Then (IN 3 .5, and N > 0 since 3:0 > .5. By (3.22), {EA/+1, S C,c ' IL‘N for all k 2 0. (3.23) cm0 Fix a positive integer m0 which satisfies MI 3 .5. For m and J, with m 2 m0, define the functions F J," by PM = Z X‘" K,,,,,. (3.24) n=-J We will now show that the functions F J,” lie in Lm. 38 It suffices to show, for each m 2 m0, that 00 Z IAI‘"IIK..,...IID, < 00. n=N Fix m _>_ m0. For each n 2 N, we have —n —n 00 1 I/\I IIKannIIDo : IAI Z’EImnIAk \kzm _ °° 1 S W A Z -IC”‘N$NI2" \k2mk °° 1 s IAI‘"c‘"‘”’"‘ z—IANIF kzmk —Nm cm n = C W IIKxNvaIDO cm A = Const — (III) S Const (.5)". Therefore, for each n 2 N, IAI‘" I|K.1..,m||1>0 S Const (5)". and so the series for FJ," converges in Lm. Claim 2: For some integer m1, greater than or equal to m0, there is a constant 6 > 0 so that IIFJJniIIDo 2 (I for all J > 0. 39 The proof of this claim is of some length, and for the reader’s reference, we note that it will be completed at statement (3.34). By (3.19), E (1 — 2,) < oo. (3.25) ng-i . 1.960 We define the function f in H °°(D) by the formula f(Z)= (1— Z)2 '11 031(‘2 (Z 6 D), "ifia where 01,, denotes the familiar automorphism of D which transposes 0 and 23,. F un- damental theory concerning Blaschke products provides that f(x,)=o 4:» 0¢ng—1. (3.26) Certainly [fl 3 4, and since {23), : 0 aé k S N - 1} Q (.5,1), it follows that [f’l is bounded (this is essentially problem #18 from [8, p. 318]). We will now prove this. Since the product defining f converges uniformly on compact subsets of D, the product rule for differentiation shows that, for any 2 E D, lf'(Z)| S 2|1-z|+|1—z|2 Z I02,IZ)|- ng—i k¢0 Hence, I 1 '— Z 2 |f( ()2: <4 + Z I (1— 12,). (3.27) k .5 and ( 1‘” ) is a decreasing l—zkz function on the interval [1,1]. From (3.27), we Obtain |f'(Z)| S 4 + 6 Z (1 - xi). (3-28) ng—i k¢0 and by (3.25), the right hand side Of (3.28) is finite. This establishes the existence Of a number B1 satisfying |f'(Z)I S 81 (z e D). Since If I _<_ 4 and | f’ | _<_ Bl it follows, in a straight forward manner, that IIfKJJOImIIDo S (4 "I" BI)IIK:ro,mII’D0 S (4 ‘I" BlIIIKxollDo for all m. We abbreviate this: IIfKIoImIIDo S 32 (m 2 1) and Observe, then, that f K mm E L, since the appropriate Taylor coefficients vanish. Since fK,0,m E Lm, IIFJ,m||vo _>_ I I/IIfKZOJTtIIDo > I Z ’\_n < foo,mIKxn,m > I/B2 nz-K 41 = I f A-"f.Z E; (Imam IIK..,..II%. —4 )3 IAI K..,m(a=.)). (3.30) n=N Because of the infinite set of zeros were were able to prescribe for the function f, the right hand side of (3.30) is independent Of J. To prove Claim 2, it suffices to find just one value of m for which the right hand side of inequality (3.30) is positive. Observe that 00 00 00 k k Z W” Iowan) = 2 MI” )3 ”—3? n=N n=N k=m oo oo (Bk (cn—N $N)k S [AI—n 0 11%;)! It; k 00 _ oo (Cn—N $0 x~)k = N ” fig kgm k 00 (C—N $0 $N)k oo ck n 2 Z k E T k=m nzN I I 00 N k 5—h- N m = E (C A” TN) (IAI) , (since — < .5) k=m k (1 — [CA—l) IAI k=m (1 Kl) _ 1 °° (1170 11m)" 1 IAIN :43... k (1 If.) |/\ [m M8 ”I? O \ [\D T;- 42 That is, ZIAI‘" K,().....x.. so .2 “/2 n=N where C is a constant independent of m. Employing this estimate in (3.30), we Obtain V 1 A 00 :13 2), IIlelvo - 2 (If(xo>IIIK..,mII%.—4Cz ————I 0; I ) k=m H — 1 00 [370]”: 00 (mo/2)]: — 32 (If($o)I Z T—AC 1.2T) 1 k=m _ °° IIISEOIIIJEII'IA‘4C'Iflb‘olk/2'c — “33.13,. A _ 1 °° (molkIIfIHSOIIISEolk—4C/2kl - 32.2. A - That is, 00 $0 _ k IIFJ,mIIDo_—B12kzm($ )kIIIf )Igfok) 40/2 I. (3.31) The comments proceeding inequality (3.30) explain why it suffices to show that the series in (3.31) is positive for just a single value of m, m 2 me. For this end, it is enough to verify that the condition [f(xo)| (:1:0)’c — 4C/2“ > 0 (3.32) holds for all k sufficiently large. Condition (3.32) is equivalent to (2x0)’° — > 0, (3.33) and since (130 > .5 we have (2:120) > 1. Hence Claim 2 is proven: for some 6 > 0, IIFJ,m1IIDo Z <5 (J > 0)- (3-34) 43 By Claim 1,- —Jo ——n 00 2: II’\ KanniIIInnl S Z3I’\In1I4\/—7I nz—oo n=Jo s M Z (/IAI"n (\/I—’\I) n=Jo _<_ Const Z (\/|—A—|)n ano < 00. Therefore, F“ i ERA “W = 133F141 is well defined in Lm,. Furthermore, “Fm,” Z 6, by Claim 2. Now we may readily complete the proof of the theorem. Using (3.16), 01:11 (le) : 07:11 ( Z X—n Kxnsml) n=—oo m — n = 2 A K$n+11ml fl=—OO 0° — n+1 = Z ’\ Kmmmi nz-oo = AF, Therefore A E 0(C, : Lm,—>Lm,). The remarks made at the beginning of the proof provide, then, that A E 0(C, : D0—>Do). By the way A was chosen, we have D\{O} g 0(0, : 1),—.00). (3.35) 44 Since “C,“ S 1 bounds the spectral radius, and since the spectrum is a closed set, (3.35) implies that 0(C, : Do—2Do) = D. /// 3.11 Theorem. If the operator C, : D0—2D0 is continuous and (C,)" is compact for some n, then (,0 has an attractive fixed point a E D and 0(C,) Q {(,0'(a)’c : k = 1,2,3, . . .} U {0}. PROOF. Fix n so that (C,)" is compact. Suppose that (p fixes no point in D. By The Grand Iteration Theorem ([9, p. 78]) (p, and consequently (,0, (the composition of (,0 with itself n times), has a fixed point in 6D at which the angular derivative exists. Thus (C,)" : H 2—->H 2 is not compact (a contradiction since (C,)" : D0——>D0 is compact). SO (0, and consequently (0,, has a fixed point in D. We shall denote it by a. (,0 is certainly a non-automorphism, and so |(p’(a)| < 1. By KOnig’s theorem ([9, p. 93]), 0(C,n) C {(p;,(a)k: k=1,2,3,...} U {0} = {(p'(a)"k: k: 1,2,3,...} U {0}. 45 As C,,, = (C,)”, the Spectral Mapping Theorem then implies that 6(C,)" g {(0'(a)"’°: k =1,2,3,...} U {0}. (3.36) Set A, = {(,0'(a)kA: A’" :1; k =1,2,3,...} U {0}, for m = n and m = n + 1. By (3.36), 0(C,) Q A,. Since (C,)"+1 is also compact on Do, the same reasoning shows that 0(C‘p) g An+1. Hence 0(C,) Q A, (I A,“ = {Ip'(a)k: k = 1,2,3, . . .} u {0}, which is the desired conclusion. A /// Remark. If (,0 : D—>D is a hyperbolic map with no boundary fixed point, or is a loxodromic map, then 0(C, : D0—>D0) = {(p'(a.)" : n = 1,2,3, . . .} U {0} (3.37) where a denotes the point of D fixed by (0. Theorem 3.11 shows that the left hand side of (3.37) is contained in the right hand side, and it is not difficult to show that each of the non-zero members of the right hand side is an eigenvalue of C,. The spectrum is a closed set, and so (3.37) follows. Furnished below is a summary Of the spectra of composition Operators on Do 46 induced by the linear fractional transformations, which are self-maps Of D. Where a appears below, it denotes the point Of D fixed by (p. e If (0 is a parabolic or hyperbolic automorphism, then 0(C,) = T. e If (p is an elliptic automorphism, then 0(C,) = {(,0’(a)" : n = 1, 2, 3, . . .} Q T. e If (0 is a parabolic non-automorphism, then 0(C,) = {€th : t E [0, oo) } U {0} for some point '11) E 11“. e If (,0 is a hyperbolic non-automorphism without a fixed point in D, then 0(C,) = D. e If (0 is hyperbolic with an interior and a boundary fixed point, then o(C,) = D. e If (,0 is a hyperbolic with no boundary fixed point, or is a loxodromic map, then o(C,) = {(p’(a)": n = 1,2,3, . . .} U {0}. Remark. Since an Operator C, : D—>D leaves the constant functions fixed, upon writing D = C CD D0, Lemma 3.9 shows that 0(C, : D—>D) = 0(C, : D0—>D0) U {1}. In particular, all of the results listed above hold for C, : D——)D, if one merely includes the point {1} in the last result. CHAPTER 4 Hilbert-Schmidt Composition Operators and Capacity Let H denote a Hilbert space. A linear operator T : H —>H is said to be Hilbert- Schmidt if : IITIBnIIIII < 00 n=1 for an (or equivalently, any) orthonormal basis {e,,}$,°:1 of H. 4.1 Theorem. ([9, p. 25]) If T : H —>H is Hilbert-Schmidt, then T is a compact operator. OUTLINE OF PROOF. For n E N, define T, on H so that T,(f) is the projection Of T(f) into LS({T(61), T(eg), . . . , T(e,)}). HOlder’s inequality shows that T,—>T as n—>oo. Since each T, is a finite rank operator, T is therefore compact. /// Denoting D by D1 here, we that C, : D,—>D,, is Hilbert—Schmidt for a E (0,1] 47 48 provided °° 2" °° IIP"I|% 2:110 (7) It. = Z—-.—° oo. e Kfi(n) x $17; as n—>oo. Let K be one of the kernels above. Let E Q T be a closed subset, and let M +(E) denote the class Of positive measures supported by E. Li(T) will denote the subset of positive functions of L2(T). We define four different capacities of the set E with respect to the kernel K: cK,1(E) = sup{||11[|: ,u E M+(E); Vt E E, K =1: 11(t) $1}, CK,2(E) = SUPIIIHIII H E M+(E); IIK * M“; S 1}, CK,2(E) = inf{||F||§: F e Li(T); Vt e E, K * F(t) 2 1}. 50 If E C T is not closed, and C denotes one of the capacity functions above, define C(E) = sup C(F) FCE where the supremum is taken over all closed subsets F . By these definitions, each of these capacity functions is defined for every subset of T and is inner-regular. In each case, it is easy to see that K13 K2 :> CK,(E) 2 CK,(E) and E1 E E2 => CK(E1) S CK(E2)- If K = K109, we sometimes substitute “log” in place of “K” in the capacity notation. [5] is a good source of information on the capacity CK’l and its relationship with trigonometric series. [6] is a good source of information on capacity functions induced by potentials—including the ones which have been defined above. In the sequel, some of our theorems will express results in terms of the big-C capacity functions CK,2. The following theorem recognizes the equivalence of these capacities with the classical ones. By equivalence, we mean that they share the same null-sets. 4.2 Lemma (Capacity Equivalence). For all subsets E Q T: (a) C,,,(E) = 0 (=> CK ,2(E) :0 i (b) CK,_m1(E) = 0 <=> CK1_%,2(E) = 0 (a E (0,1)) 51 The proof will use results from both [6] and [5]. The following identity is from [6, p. 273] and holds for all compact subsets E Q T: Nit- CK,2(E) 1' (CK’2(E)) . (4.1) Combining Theorems III and V from [5, pp. 37,40], we Obtain the following lemma: 4.3 Lemma. There exists 0 75 ,u E M +(E) satisfying 00 I e 2 00 A 2 MI") | |u(n) | < 00 01‘ < 00 E... InI .33.. In 11-“ 119150 n¢0 iff clog,1(E) > 0 or cKm1(E) > 0, respectively. PROOF OF LEMMA 4.2. For the purposes of this proof, define K0 : K109. Then statement (a) is statement (b) with a = 1. Thus it suffices to prove (b) for arbitrary a E (0,1]. Fix such a number (2. Since these capacities are inner-regular, we may assume that E Q T is a compact subset. Claim 1: CK,_,,2(E) > 0 => CK1_,,,1(E) > 0. Suppose CK1_9,2(E) > 0. Then by (4.1), cKl_%,2(E) > 0. By the definition Of CK1_%’2, there exists 0 74 11 E M +(E) satisfying IIKl—g * #II2 < 00- 52 Then we have 00 > ”Kl—gwili 1 = -— K_g* 2dm 27r.[rI 12 III . oo = Z I (Kl—a, *MIAIT’A) I2 nz—oo 0° 2 2 = Z I(K1—%) (n)I |#(n)I - n=—oo Hence we Obtain 11750 f: IMn) I2 .=_.. INI“ n¢0 Lemma 4.3 implies then that CK,_,,1(E) > 0, completing the proof of Claim 1. Claim 2: cK,_m1(E) > 0 => CK1_,,1(E) > 0. Suppose cK,_m1(E) > 0. Then by Lemma 4.3, there exists 0 75 11 E M +(E) satisfying 00 e 2 Z I 11(7).) I < ..=_... In I“ 11.960 Hence 00 A 2 Z IHI?) I < 00, .z-.. WI2 11950 and this shows that K145. =1: 11 E L2(T). Considering the definition of cK1_ 312’ this im- plies that cK1_g,2(E) > 0. Using the identity in (4.1), it follows that CK1_%,2(E) > 0. This completes the proof Of Claim 2 which, along with Claim 1, completes the proof 53 of the theorem. /// By Lemma 4.2, if some property occurs capacitarily almost everywhere (i.e. except on a set of capacity zero) with respect to one of these capacities, then it occurs capac- itarily almost everywhere with respect to the corresponding capacity (as indicated by Lemma 4.2). We frequently use the abbreviation CK, — a.e. e” E T, et. al., to mean capacitarily almost every member of T with respect to the capacity CK,j (j = 1 or 2). 4.4 Lemma (Weak Capacitary Inequality). Let K be a kernel and F E L1(T). For a > 0, set E, = {e“ : K * F(e”) 2 0.}. Then IlFl|§_ CK,2(Ea) S a2 PROOF. By definition, CK,2(E,) = inf{||F||§ : F E Li(T); Ve” E E,, K * F(e“) Z 1}. Therefore, since K :1: ac“) Z 1 for each e” E B, we have IIFI|§_ a2 CK.2(Ea) S IIF/a||§ = /// 54 Remark. For F E L2(T), K >1: F is certainly defined pointwise wherever K * |F| is finite. Lemma 4.4 then shows that K at F(e“) is defined (and finite) for CK; — a.e. e“ E T. 4.5 Lemma. Fix a E (0,1] and define K = K1_%. Then for each f E D,, (where D1 = D), there exists F E L2(T) satisfying oo f(2) = 2(K * F)A(n) z" (z E D); (4.2) n=0 moreover, for f and F associated in this way, IIfIli), >< IIFIIE (f E Do.)- (43) PROOF. Recall that K(n) > 0 for each n E Z, and that K(n) x :1, as n—mo. Let f E D,,. Define a sequence c = {c,,}§,’°=_co by (n E Z). Then [C,,]2 x n“ | f(n) I2 as n—>oo. Since f E D,,, it follows that c E l2(n). By the Riesz-Fischer theorem, there exists F E L2(T) with F (n) = C,, for all n E Z, and this gives (4.2). Since n"‘(K(n))2 ><1 for all n E Z \{O} 55 and A (K(0))2 x 1 (trivially), the implicit pairs of constants associated with each of these statements can be chosen to be the same. Then for any f E D,,, we have IIfIIAD, = I f(O) I2 +27%“ | f(n) |2 n=1 = I (K . F)‘(0) l2 + in“ I (K . mm |2 11:1 = I Rm) 121 Pm) I2 + i n“(K(n))2 I 1%) l2 )( 1|13“(0)|2+§_A.11|15‘(n)|2 = I|F||§- We note that the implicit constants associated with x here are the same as those we considered above (and are independent of f E D,,). This yields (4.3). /// The following two theorems, which are well known, help substantiate the state- ment make in the introduction that capacity tends to play the role in the Dirichlet and Dirichlet-type spaces that Lebesgue measure plays in H 2. They show that func- tions in these spaces have boundary values and Lebesgue points capacitarily almost everywhere. 56 4.6 Theorem. Let f E D, with a E (0,1] (where D1 = D). Set K = K1413. Then the limit f(e“) 21133163) exists (and is finite) for CK, — a.e. e” E T. Remark. By Lemma 4.2, CK; may be replaced above by C,,, if f E D, or by c1_,,1 if f E D,,. PROOF. Fix f,K and a as in the statement of the theorem. By Lemma 4.5, there exists F E L2(T) satisfying f(2) = :(K =1: F)A(n) z" (z E D). By the Remark following Lemma 4.4, K =1: F (ei‘) is defined for CK, — a.e. e” E T. Define Q for all such points by {2(6“) 2 lim sup [f(re“) — K * F(e”)|. r—+1— Hence C(e“) = lim sup |P, * K I: F(e”) — K * F(e“’)|. (4.4) r—rl' Let c > 0. For h E C(T), define g=g(h) iF—hEL2(T). 57 Then F = g + h and for all e“ E T, P, =1: K :1: h(e”)—-)K =1: h(e”) as r—21‘ . Therefore, (4.4) becomes 52(6”) = lim sup IP, :1: K =1: g(e”) — K =I< g(e”)| r—+1" 3 lim suplP, * K =1: g(e”)| + [K * g(e“)|. r—rl“ Observe that P, * K at g(e”) = K * P, * g(e”) 1 2.. . = —/ K(t—0) P, *g(e”)d0, 2K 0 SO - it 1 2" 1‘9 limsupIP,*K*g(e )l _ 27?].1 K(t—6)M,,,(P[g])(e )d0 r—rl— : K* MradIPIQIXeitla where Mm, denotes the radial maximal function. Therefore, 9(6”) S K * Mmd(PIgl)(e“) + |K * g(€“)|- We denote the Hardy-Littlewood maximal function by M HL. By Theorem 11.20 Of I8], 9(6“) S K * MHLIQIIBII) + IK * 9(61IIIa 58 and this easily implies that 0(6”) S 2 K * MHL(9)(6”)- (4-5) By inequality (4.5), {eitz Q(e”) > 6} Q [6”: K at MHL(g)(e”) > 6/2}. Hence by Lemma 4.4, CK,2({e“ : 9(e”) > 6}) l/\ CK,2({6“ 3 K * MHL(9)(eit) > 5/2” (2/5)2 IIMHL(9)II2 3 (2M2 IIMHLII2II9113. (4.6) |/\ Recall that g 2: F — h where h was as arbitrary continuous function on T. Since the continuous functions are dense in L2(T), we may choose 9 = g(h) and It so that [Igllg is as small as we please. Therefore, the inequality above may be improved: CK,2({e” : 9(e”) > 6}) = 0. (4.7) Note that (4.7) holds for each 6 > 0. Using the a-subadditivity of 0K3, we Obtain CK,2({e“: 9(e”) > 0}) = 0. Considering the definition of (2, this completes the proof Of the theorem. /// 59 4.7 Corollary of the proof. For f, K and F as in Lemma 4.5, f(e”) = K =1: F(e”) for CK; — a.e. e” E T. 4.8 Theorem. Let f E D,, with a E (0,1] (where D, = D). Set K 2 K1_%. Then CKg—ae e” E T is a Lebesgue point of f. The proof Of this theorem is analogous to that Of Theorem 4.6. OUTLINE OF PROOF. We identify t with e” E T. Let f, g and h be as in the proof of Theorem 4.6. Then K * F = f (CK; — a.e.) and K =1: h is continuous. Define Q by be“) =11msupi /, mo) — WM 40 r—+0 7' -r for each t where f(t) is defined (CK; — a.e. by Theorem 4.6). Hence for CK,2 — a.e. e” E T, . 1 t-i-r 9(e”) = limsupE/ [K * 9(6) — K at g(t)[ d0 3 MHL(K * g)(t) + [K =I< g(t)|. r—>0 Note that 1 Hr M...(K *9)(t) = sup (,— / IK *QIledy) r>0 T t-r SUD (511; [If i(11%) Ig(y - 8)| ds dy) r>0 —r 271' |/\ 60 S 51-7,]: K( (8) Egg/H lg(y-S)|dy) ds 3 5,1136) MHL(9)(t—3) ds = K*MHL(g)(t). Hence we see that for CK; — a.e. e” E T, Q(8“) S K * M111.(.¢1)(I)+IK * 9(t)I < 2 K * MHL(g)(t). The remainder Of the proof is the same as that of Theorem 4.6. /// 4.9 Lemma. If a < 2 and h is the function defined by = Z n1 7121 then 1 2—0 h(:1:) (1_ x) as (IS—>1 PROOF. If a = 1, very little analysis is required to Obtain the result. Therefore assume 1 79 a < 2 and define 9(2) = (-I—)2_a. Then 9(0) = 1 and, for all n 2 1, 1—1: gI")(0)=(2—a)(3~—a)-~(n+1—~a). (4.8) 61 Observe that §(n) > O for all n 2 O, and that h(n) > 0 for all n 2 1 (here £702) and h(n) denote the 12‘” Taylor series coefficients of g and h). It suffices then, to show that h(n) x “(12) as n——>oo. By (4.8), for n 2 1, M _ n! nl’a Mn) _ 9"”(0) n! nl'a Z (2—oz)(3—a)---(n+1—a) n! nl‘a = (1 — a) (4.9) (1—a)(1—a+1)(1—a+2)-~(1—a+n)’ Consider the following formula, due to Gauss [11, p. 312]: nlnz 'lqutloz(z+1)~-(z+n)—P(z) (260). Its application to (4.9) shows that ) Jigfl- = (1 —a)I‘(1——a). (71) Ca) In particular, this limit exists and is non-zero; this shows that h(n) >< 9(n) as n—>oo. /// In Theorem 3.1 of [10], Shapiro gives a short direct proof that the condition 211‘ dt ——+— < oo 4.10 f. 1— Mew ( ’ is both necessary and sufficient for a self map

0, as §—> 1’. (4.11) Shapiro’s proof that (4.10) is necessary for Hilbert-Schmidt composition operators on 62 H 2 provides orientation for the proof of the following theorem concerning the Dirichlet and Dirichlet-type spaces. Hansson’s Inequality (cited below), and Theorem 4.6 and its corollary are important ingredients of its proof. One will find it interesting to compare (4.11), satisfied when Cw is Hilbert-Schmidt on H2, with Corollary 4.11 concerning Hilbert-Schmidt composition operators on D and Do. 4.10 Theorem. Fix a 6 (0,1] and define K = K 1-05. If the composition operator C,,, : D,,—>730 is Hilbert-Schmidt (where D1 = D), then [01 CK,2({eit3 I‘P(e“)l Z 5}) (16%?” < 00- PROOF. cp" is in Do, for each n E N. By the Corollary of the proof of Theorem 4.6, for each n E N there exists E, E L2(T) with ABE—Hg) LID" n=1 n 2 i '5‘ [01 CK,2({e“= we“) 2 ownswc) 11:1 1 , . °° = 2 [0 were“: 19469123) 2121-08" as. n=l A00 CK,2({€“Z K * IFnKeit) Z A}) d(A2) > (.1 Came“ = )K * F.(e“)1 2 A)) d(X") (4.12) The only possible singularity of the integrand in (4.12) occurs at E = 1, so Lemma 4.9 implies that 1 00 > [0 ems“: lso(e“)l2€})(-1—_—,C.2)2—_a dg. 64 This quickly yields the result stated in the theorem: [010K2({€it1 |90(€“)| 2 €}) (1%)?3 < 00- /// 4.11 Corollary. Fix a 6 (0,1] and define K 2 K1_%. For a self map (,0 of the disk, define the capacitary distribution function W) i CK,2({6“ = |90(e‘3)| Z t})- If GP : D—>D is Hilbert-Schmidt, then there exists a constant M satisfying If C,,, : D,,—+130, is Hilbert-Schmidt and a 6 (0,1), then there exists a constant M satisfying PROOF. By Theorem 4.10, there exists a number M1 satisfying 614 (fl—C)?“ for all t 6 [0,1). (4-13) M12 [gm 65 If a = 1, then define h(() = log 1—_1_—C; if a 76 1, then define h(() = 53(1— 0““. Then 1 h'(C) = W Fix t E [0, 1). Inequality (4.13) and integration by parts shows that M. 2 homo): — [O‘h(<)dg(<) = g(t)h(t)—g(0)h(0) + [handym- Hence, M2 2 M1 + 9(0) [1(0) > g(t) h(t), and we obtain M2 Z 9“) Mt)- If a = 1, then this gives the desired result with M = M2. If a 79 1, then this gives the desired result with M = (1 — a)M2. /// Remark. From Theorem 4.10 (or Corollary 4.11), and Lemma 4.2, we see that Czog.l({e“= |’D or C,,, : Da—)Da is Hilbert-Schmidt, respectively. BIBLIOGRAPHY BIBLIOGRAPHY [1] C. C. Cowen and B. D. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Raton, 1995. [2] C. C. Cowen, “Composition Operators on H 2”, J. Operator Th., 9 (1983), 77— 106. [3] K. Hansson, “Imbedding Theorems of Sobolev Type In Potential Theory”, Math. Scand., 45 (1979), 77—102. [4] E. Hille and R. S. Phillips, Functional Analysis and Semigroups, revised ed., American Math. Society, 1957. [5] J. P. Kahane and R. Salem, Ensembles Parfaits et Séries Trigonométriques, Hermann, Paris, 1963. [6] N. G. Meyers, “A Theory of Capacities for Potentials of Functions in Lebesgue Classes”, Math. Scand. 26 (1970), 255-292. [7] E. A. Nordgren, “Composition Operators”, Canad. J. Math., 20 (1968), 442- 449. [8] W. Rudin, Real and Complex Analysis, third ed., McGraw Hill, New York, 1987. [9] J. H. Shapiro, Composition Operators and Classical Function Theory, Springer- Verlag, New York, 1993. [10] J. H. Shapiro, “Compact, Nuclear, and Hilbert-Schmidt Composition Operators on H 2”, Indiana Univ. Math. J ., 23 (1973), 471-496. [11] R. A. Silverman, Introductory Complex Analysis, Dover Pub., New York, 1972. 66 "11111))111111))“