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(:- -.. .. viii-L 0, .. ~33 ‘».,- .v...’ "fr ... & “up, L,.,. i31.“.5’:"{ 1:3" #54,, _ _ L5§DLJC¥ ‘ I III!" $37227 ‘m-uu. y) . m u 3 - I 4 Art‘u‘ém . 3., ..., };»'=5 ¢ ', ‘ l, ‘ mm 1;! ...L ‘. “13.243. : [MICHIGAN sure u ll m ll I'll/l ”ll/IT WWII/l m7 II/ II {mi/.0015; W This is to certify that the dissertation entitled Dirichlet Spaces on finitely connected domains presented by Young—Chae Nah has been accepted towards fulfillment of the requirements for Pb. n degree in Mathematics. Major professor Date W [9, ’9?! MS U is an Affirmative Action/Equal Opportunity Institution 0-12771 IBMRY Michigan State University v“ \_ w PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES mm on or before date due. DATE DUE DATE DUE DATE DUE D E ! ix" 1 MSU le An Affirmative ActiorvKuel Opportunity Institution I ,, °m"“?'“' DIRICHLET SPACES ON FINITELY CONNECTED DOMAINS By Young-Chae Nah A DISSERTATION Submitted to Michigan State University in partial fufillment of the requirements for the degree of DOCI‘OR OF PHILOSOPHY Department of Mathematics 1991 é37”fJ5/‘0 ABSTRACT DIRICHLET SPACES ON FINITELY CONNECTED DOMAINS By Young-Chae Nah Suppose Q is a finitely connected nonempty domain in C such that no connected component of an is equal to a point. In the second chapter we show that o¢(M¢), the essential spectrum of a multiplication operator Mq, on the Dirichlet space 0(9), is equal to cl(; am when (2 has an analytic boundary. By the conformal invariance of 08(M¢) and cl(B(Q)= Lfg dA (1.1) where (M denotes the usual area measure on Q. Let 20 be in Q. The Dirichlet space D(Q,zo) is the Hilbert space of analytic functions f on (2 such that Al f '|2 M < co and flzo) = O, with the inner product 061): L f' -g—' (M. (1.2) Changing the distinguished point zo gives a space that is obtained from the original by subtracting a suitable constant from each function. We will use 0(9) instead of D(Q,zo) if the distinguished point is irrelevant. 2 The square of the Dirichlet norm of f is just the area of the image of 9 under f, counting multiplicity. . 2 We Will use Ilflle) where H(Q) is the set of analytic functions on Q. to denote I If ' |2 (IA even when f s 11(9) \D(Q,zo) 0 It is well known that point evaluation maps on 3(0) (see Conway [8], Chapter III, Corollary 10.3) and 0(9) (see Taylor [14]) are bounded. Here we will prove the boundedness of point evaluation of each derivative on D(fl). Lemma 1.3: Let 2 e 52 and let n e Nu{0}. Then the map Xusz, 20) —-> c defined by I”. (f) =sz) is a bounded linear functional. Proof: Let 25 0. First assume n=0. Wewilluse 7», instead of 1,2,0, Let I‘ bearectifiablepathin Q from 20 to 2. Then Iml=ytzn=l #01,)de s (lengthofI‘) sup{ lf'(w)l:we r}. (1.4) Since I‘ is a compact subset of 9, there exists r > 0 such that the distance between I‘ and 89 is bigger than r. Let g 6 3(9) and let B(w,r) be the open disk in C centered atthepoint w withradius r. Foreach we I‘, lg(w)l s 43 jlg< : )l dA(t) bythemeanvalue property 1|: r B(w,r) s -1— II II - - R r 8 B(Q)° by Holder's rnequahty Since f ' 6 3(9), there is a constant K, which depends only on I“ and 9, such that the right hand side of (1.4) s K Ilf'llB(a)— = K Ilf "0(a) Therefore the point evaluation map 1., is bounded. Now let n 2 1. Choose 5 > 0 such that m C 9. Here m denotes the closure of 3(z,5) in (3, Since {2.9: weBB(z,8)] is a subset of the dual space D(Q,zo)* and sup{ l lw( f ) I : we BB(z,8)} < no for all f e D(Q,zo), by the Uniform Boundedness Principle, there is a constant K such that sup{ llkwll : w e aB(z,8)} S K. Hence lf(w)l s lawn ll f "mm s K II f "mm for all fe D(Q,zo) and for all we 83(z,8), By the Cauchy Formula, It,,,(f)|=|f<">(z)l sg— ifs-ig—Z-%+—lldwl sg—ixnfnmm forall fe D(Q,zo). Q.E.D. Using the same argument as in the proof of the above lemma, we can prove that every norm bounded subset of D(Q,zo) is uniformly bounded on each compact subset of Q. In a normed vector space, every weakly convergent sequence is norm bounded. Hence we get the following lemma by the normal family argument. Lemma 1.5: Let ne Nu{0]. If {fm} isasequence in D(£2,zo) converging to f weakly, then fit") —> f ('0 uniformly on compact subsets of Q as m —) co. 4 Remarks: (a) If Va)“ A is a bounded net in D(9) such that fa converges to f weakly. then we can still apply the normal family argument to prove that fé") -> f (n) unifome on compact subsets of 9 for all n e N. (b) A weaker version of the converse of the above lemma will be discussed in Lemma 3.3. An analytic function (I) on 9 is called a multiplier of D(9,zo) if (pD(9,zo) c D(9,zo). We denote by M (D(9,Zo)) the set of all multipliers of D(9,zo). For any multiplier (p, the linear transformation MlP : D(9,zo) -> D(9,zo) defined by Mq, f = (pf is bounded; this follows from the Closed Graph Theorem and the boundedness of point evaluation maps. Mq, is called a multiplication operator. Giving each function in M (D (9.20)) the operator norm of the corresponding multiplication operator makes M(D(9,zo)) into anormed space. Standard references for M(D(U)) are [13] and [14]; here U denotes the open unit disk in C. If (p e M(D(9,zo)), then (p is in the set of bounded analytic functions H°° (9) with H (p II“, S II 114.," (see [10], Lemma 11). But the converse is not true. Actually M(D(U)) is not even closed under ll - IL, norm (see Axler and Shields [5], Theorem 10). Lemma 1.6: If 9 is bounded, then (p' e 3(9) and tp— 0 such that 3(20, r) c 9. Then co > I lz-zolzltp'lsz > f lz-zolzlrp'l2 dA + r2 j |(p'|2 dA. 9 3(20, r) 9\B(zo, r) Hence I Itp'PdA < co. Since (p' is bounded on 3(20, r), (p' is in 3(9). Now the (“(20, I) second assertion follows immediately. Q.E.D. The following lemma can be proved using change-of-variables. Lemma 1.7: Let 9, and 92 be two domains in C and let zoe 91 and wo e 92. Suppose \v is a conformal mapping from 92 onto 91 such that \y(wo) = 20. Then (1) The composition map CV: D(91.zo) -) D(92,wo) defined by CV0) =f-\|I is a unitary map. (2) The composition map CV:M(D(91.zo)) -+ M(D(92,wo)) defined by Cw(q>) = (pow is an onto isometry. For the open unit disk U in C, the spaces D(U,O) and 3(0) can be described in terms of Taylor coefficients using (1.1) and (1.2); namely flinging)”; n Ianlz, (1.8) .. l I2 n f '53!) = 1: 3:30 :21 . (1.9) where flz) =n§oanz". Hence we have D(U,O) c 3(U). For a simply connected domain 9, there are some equivalent conditions to D(9) c 3(9) (see Axler and Shields [5], Theorem 1). 6 Denote the annular region in C centered at 0 with the inner radius r > O and outer radius 1 by 11,. Suppose 1(2) = 2‘, anz". Then "jug“ ), nfufiu ) can be W r r written in terms of the Laurent series coefficients similar to those which are in (1.8) and (1.9). The formulae are given in the following lemma, which can be proved by direct calculation. Lemma 1.10: Suppose flz) = E anz". rpm (1) If fe D(A,,\lr ), then IIfII12,(Ar)= n "Eon IanZ (1- r 2n). (1.11) (2) If fe 301,), then la 12 ":1 (1-r2n+2) - 2n Ia_1|2 log r. (1.12) lung“), = 1: "£1 If the infinite series (1.11) converges, then so does the series (1.12). Hence we have D(Arnl—r) c 3(Ar). Now, by the Closed Graph Theorem, the inclusion map I : D(Ar) —> 3(A,) is bounded. The next lemma establishes some equivalent conditions for D(9) c 3(9) when 9 is a bounded doubly connected domain in C. These results can be proved as Theorem 1 in [5]. Recall that any doubly connected domain is conformally equivalent to some annulus (see, for example, Axler [1], Doubly Connected Mapping Theorem on page 255). 7 Lemma 1.13: Let 9 be a bounded doubly connected domain in C, and let 20 e 9. Suppose 1|! is a conformal mapping from A r onto 9 for some r > 0. Then the following are equivalent. (1) man) c 8(a) (2) 2 061.20) c 0(on (3) w D(Ar. w'leo» c D(Ar.\ll'1(lo)) (4) w'D(Ar.\v'1(zo)) c 3(Ar) Remarks: (a) (1) and (2) are equivalent for any bounded domain 9. (b) (3) and (4) are equivalent for any bounded domain 9; namely if \u e H (9) where 9 is a bounded domain, the two statements 1)! e M(D(9)) and \y‘D(9) c 3(9) are equivalent . (c) Suppose 9 in the previous lemma has an analytic boundary; namely 9 has two analytic curves as a boundary. Then \y can be extended analytically up to 8.4, by Schwarz Reflection Principle. (For the definition of an analytic curve and the analytic extension of w, see page 12 of this thesis.) Hence lw'l is bounded on A r and so condition (4) in the above lemma is satisfied. Here we used the fact that 001,017) c 3(A,). Therefore we have D(9) c 3(9), z is in M(D(9,zo)), and especially ‘I’ e M(D(Ar». In Chapter 2, we will see that D(9) c 3(9) when 9 is a finitely connected domain with an analytic boundary. (d) For any domain 9, if D(9) c 3(9), then the inclusion map I : D(9)—)3(9) is bounded by the Closed Graph Theorem 8 We do not know exactly when the inclusion map I :D(9) c 3(9) is compact. For simply connected domains, Axler and Shields got some results (see Axler and Shields [5]). For doubly connected domains, we have the following lemma. Lemma 1.14: Let 9 be a doubly connected domain and let \(t be a conformal mapping from A, onto 9 for some r > 0. Then the following are equivalent. (1) The inclusion map I : D(9) —) 3(9) is compact. (2) The multiplication operator Mv' :D(A,) -> 3(A,) defined by M v'( f ) = wf is compact. Proof: Define an operator T :3(9) -> 3(A,) by T(g) = \v'(g-\|I). Then, by change-of-variables, T is isometry . Let h e 301,). It is easy to see that ($17 h )out'1 is the preimage of h under T, again by changeeof-variables. Hence T is a unitary map. Note that Mv' - CV = T - I on D(9) where CV is the composition map as in Lemma 1.7. Since CW and T areboth unitary, (l) is equivalent to (2). Q.E.D. We proved that a point evaluation map on the Dirichlet space is bounded in Lemma 1.3. When 9 is either U or A,, we can find A, explicitly by direct calculation. Suppose 9 = U and z e U. Then 2», defined by l A, (w) = % log (1.15) 1 -?w is the point evaluation map at 2 on D(U,0). When 9=Ar and zeAr, h, definedby =1 rho/7)" _ n Mw) “€0,104,” M (117)] (1.16) is the point evaluation map at z on D( Aral—1:). It does not seem possible to express the infinite sum in (1.16) in closed form. CHAPTER 2 ESSENTIAL SPECTRUM OF MULTIPLICATION OPERATORS Recall that an operator T on a Hilbert space H is called Fredholm if the kernel of T and H/TH are both finite dimensional vector spaces. These conditions imply that T has closed range (see [6], Cor 3.2.5). Suppose T is an operator on a Hilbert space H. The essential spectrum of T, denoted 03(7), is defined to be the set of complex numbers c such that T - c is not Fredholm. oe(T) is precisely the spectrum of T in the Calkin algebra L(H)/K (H) where L(H) denotes the set of all bounded operators on H, and K (H) denotes the set of all compact operators on H (see Douglas [9]). If (p is an analytic function on 9, then the cluster set of (p on an, denoted cl(q);39), is the set of complex numbers c such that there exists a sequence {2"} in 9 such that 2,, tends to 39 and f(z,,)—)c as n—ioo. Suppose G is any open set in the complex plane C such that no connected component of 80 is equal to a point. On the Bergman space 3(G), Sheldon Axler showed that oe(Mq,) = cl(q>;BG) when q>e M(3(G)) (see Axler [2], Theorem 23). No result of this generality is known for the Dirichlet space. If 9 is a bounded simply connected domain, and if (p is a multiplier of D(9), then oe(Mq,) = cl(q>;BG) (see Axler and Shields [5], Theorem 11). In this chapter, we will prove that the same conclusion holds when 9 is a bounded finitely connected domain (with an analytic boundary) and (p is a multiplier of D(9). 10 11 Suppose 9 is bounded. If Mq, : D(9,zo) —> D(9,zo) is a multiplication operator and oe(Mq,) = cl(tp;a9), then Mq, is also a multiplication operator on D(9,zl) for any zle 9 and has the same essential spectrum; namely the essential spectrum of a multiplication Operator on the Dirichlet space does not depend on the choice of the distinguished point. Furthermore the essential spectrum of a multiplication operator is conformally invariant in the following sense. Lemma 2.1: Suppose 91 and 92 are domains in C and ‘l’ is a conformal mapping from 91 onto 92 such that 111(20) = wo. Suppose (p e M(D(92,w0)) and oe(Mq,) = cl(q>;a9,). Then (pow e M(D(9l,zo)) and 0e(quy) = cl( 891 and q>-\|I(z,,) —) A as n —) 00. Since {w(z,,)} is a sequence in 92, l2 and 11t(z,,) —) 89») (maybe some subsequence of {w(zn)}). A. e c1((p;892). Hence cl(tp-\1t;89l) c: cl(q>;a9,). The other inclusion can be proved similarly. Q.E.D. By an analytic curve, we mean the image of the unit circle in C under a one-to- one function analytic on a neighbourhood of the unit circle. Lemma 2.5: Suppose 9 is a bounded doubly connected domain with an analytic boundary. Let t)! be a conformal mapping from 9 onto A , for some r. Then w and 1|! ‘1 can be extended analytically up to 39 and 8A, respectively. Proof: A well known extension of Jordan Curve Theorem (see, for example, Koosis [11], page 53) says that \II has a continuous and one-to-one extension up to 39. Suppose 89 = TOUT] where To is the boundary of the unbounded component of Sz\ 9. Since 1‘1 is an analytic curve, there exists a neighbourhood N1 of EU and a one-to-one analytic function (p1 on N1 such that (pl(aU) = I}. We may assume that q>1(UnN1) isin 9. Note that wool is analytic on Uan and continuous on 77an. Since Nt-(p1(z)l —) r as z —) 3U, by the Schwarz Reflection Principle, \|t-(p1 has an analytic extension up to EU. Suppose N1' is a neighbourhood of EU on which \lt-tpl is analytic. Then \lt-tpl-tpl'l is an analytic extension of \v on q)(N1nN1'). Similarly we can extend ‘l’ analytically up to To. Note that (pl'low '1 is analytic on a neighbourhood in A, of [ze C: Izl = r] and continuous up to {ze C: Izl = r}. Since ltpl'l-V '1(z)| -> 1 as Izl—i r, by the Schwarz Reflection Principle, (pl‘low '1 has an analytic extension up to {ze C: Izl = r}. Suppose N2 is a neighbourhood of {ze C: Izl = r] on which (p1‘1.\|t '1 is analytic. Choose a neighbourhood N2' of {ze C: Izl = r] such that Nz' CN2 and (pl'low '1(N2') c N1. 13 Then (p1.(pl’1.\|t '1 is an analytic extension of \lt'l on Nz'. Similarly we can extend \ll‘l analytically upto {ze C: Izl = 1}. Q.E.D. Suppose 9 is a bounded simply connected domain in C (or an interior of the complement in 82 of a bounded simply connected domain in C) with an analytic boundary. We claim that each C in 39 belongs to a closed ball contained in the complement of 9. (This condition is called the external ball condition.) Let ‘V be a conformal mapping from 9 onto U. Then, by the above lemma, ‘1’ can be extended analytically to some neighbomhood G of 9. Define a function (p on \|t(G) by tp(z) = Izl2 - 1. Then (pow is called a C” defining function of 9; namely (pow is a real-valued 6'" function on some neighbourhood 0 of 39 satisfying following three conditions: (i) 9 n 0 = [ z e 9 : (p-w(z) < 0]; (ii) 39 n 0 = { ze 9: tp-ut(z) = 0}; (iii) the gradient vector of (pout on 89 is never 0. Note that the tangent line of 39 at C is perpendicular to the gradient vector ViP'WQ- Now, to prove the claim, we may assume that 0 e 89 and the tangent line of an at O is the real axis. Then near 0 in R, an is the graph (x, A(x)) of a C” function A defined on a neighbourhood of 0 in R, where A'(O) = 0; this follows from the implicit function theorem. Since A'(O) = O, |A(x)l = 0(lxl2) as x-)O by Taylor's Theorem. Hence the external ball condition at 0 is satisfied. Actually the external ball condition is satisfied when 9 has a C2 boundary (see, for example, [4], Chapter 10). Let 9 be a bounded domain in C whose complement (in 82) consists of exactly m+1 nontrivial components where m is a positive integer. Then m+1 applications of the Riemann Mapping Theorem produce a one-to-one holomorphic mapping of 9 onto a bounded domain whose boundary consists of m+1 mutually disjoint analytic curves. Hence, as far as the essential spectrum is concerned, by Lemma 2.1 we may assume that l4 9 is a bounded domain whose boundary consists of m+1 mutually disjoint analytic curves. Let 9 be a bounded domain whose boundary consists of m+1 mutually disjoint analytic curves. The following notation is used throughout this thesis. The m+1 mutually disjoint analytic curves consisting of 89 will be denoted by T0, T1, , 1“,", where I}, is the boundary of the unbounded component of S2 \ 9. 90, or sometimes U0, will be used to denote the bounded component of S2 \ To, and U ,- will be used to denote the unbounded component of 82 U} for each j = 1, , m. And we will also denote 900 U} by 9]- foreach j=1,«- ,m. Lemma 2.6: Suppose 9 is a bounded domain in C whose boundary consists of m+1 mutually disjoint analytic curves. Let zoe 9. If f e D(9,zo). then there is a function f}- in D(9)-.20) for each j = 1, 2, , m, such that f = fo+ f1 + + fm on 9. Proof: For simplicity, we will prove this lemma when m = 2. Let 2 e 9, and let 70, 71, and 72 be mutually disjoint smooth simple closed curves in 9 so near 1‘0, 1‘1, and 1‘2 respectively that z is interior to 7]. for each j = O, 1, 2. ( 70 is oriented counterclockwise, 71 and 72 are oriented clockwise.) By the Cauchy Formula, f(z)= -—1—. J %(_c—:—d§ _1_ f(§)dC+§1,5 %:C_:_d§+§l5 J%%d§ (2.7) 70 71 72 15 We will denote the (i+1)th integral in (2.7) by gj (z) for each j =0, 1, 2. Then gj (z) is independent of the choice of y}. and is in H( U j ). Let 1;- (z) = g j (z) - gj- (20). Then fj- isin H( 91-) and 13(20) = 0 for each j. Let A0,A1, and A2 be mutually disjoint connected neighbourhoods of To , I‘l , and 1‘2 in 9 respectively. Now we will prove that lfo'l is square integrable on 90 with respect to the usual area measure dA. On 90\Ao, fo' is bounded. On A0, If 'I, | fl'l, and lf2'l are square integrable and fo' =f ' -f1' -f2'. Hence I fo'l is square integrable on 90 and so f0 is in D(9o,zo). Similarly we can prove that f1 e D(91,zo) and f2 e D(92,zo). Q.E.D. Remark: Since D(9) is conformally invariant, Lemma 2.6 is true when 9 is any finitely connected domain. Corollary 2.8: Suppose 9 is a bounded domain whose boundary consists of m+1 mutually disjoint analytic curves. Then D(9) (:3(9) and z e M(D(9)). Proof: Let f e D(9). By Lemma 2.6, f =fo+f1 +-~-+f,,, where f1 e D(9j). Since each 9,- is either a simply connected domain with an analytic boundary or a doubly connected domain with an analytic boundary, by remark (c) following Lemma 1.13, f,- is in M9,) for all j. Hence f e 3(9). Now 2 e M(D(9)) follows from remark (a) following Lemma 1.13. Q.E.D. Let P(9) be the set of polynomials. Then (1.8) shows that P(U)nD(U,O) is a dense subset of D(U,0). Also, from (1.11), we can see that n n { )3 ajzi: neNU{0},aje Cforallj=0,il,...,in, ajzai=01 (2.9) ‘ -n J=-n 1= 16 is a dense subset D( A,,zo). For a finitely connected domain 9 with an analytic boundary, we will prove, in the following theorem, that the set of rational functions in D(9,zo) whose poles are off 9 is a dense subset of D(9,zo). Let R(9) denote the set of rational functions whose poles are in sz\ 17. Theorem 2.10: Suppose 9 is a bounded domain whose boundary consists of m+1 mutually disjoint analytic curves. Let 20 be in 90. Then R(9)nD(9,zo) is a dense subset of D(9,zo). If 9 is simply connected, then P(9)nD(9,zo) is dense in D(9,zo). Proof: Let f e D(9,zo). By Lemma 2.6, there is fie D(9j,zo) for each j = 0, 1, ,m such that f=fo+fl + +fm on 9. Let 8>0. For 90, there is aconformal mapping W0 from 90 onto U such that 1|!(zo) = 0. By Lemma 1.7, fo-w'o‘ e D(U,0). Since P(U)nD(U,0) is dense in D(U,0). there is a polynomial p e P(U)nD(U,0) such that - e n fo-vg - p 10(0) ("253—13 . By Lemma 1.7, the composition map CW0 is a unitary map from D(U,0) onto D(90,zo). Hence - - g A Ilfo-p-wollbmo): "(fo-wo‘ -p)-\VOIID(QO)=II fo-wol mum!) < 5631? . (2.11) 17 Since To is an analytic curve, ‘l’o extends to be analytic to a simply connected neighbourhood G of the closure of 90 by Lemma 2.5. Hence patio is analytic on G. By Runge's Theorem, there is a sequence of polynomials {pn} that converge to p-tyo uniformly on compact subsets of G. Therefore pn' converges to (patio) uniformly on the closure of 90, and so p" converges to P'Wo in D(9O,zo). We may assume p,,(zo) = 0 for all n, by replacing pn(z) by pn(z) - pn(zo) if necessary, because the constant term does not contribute to the Dirichlet norm. Hence there is a polynomial pa in D(90,zo) such that 8 llp-Wo-po "06%) < m . By the triangle inequality and (2.11), llfo- poll ”(90)< . Note that this proves the second part of theorem. Now fix j = 1, 2, , m. Since 9]- is doubly connected, there is a conformal mapping w,- from 9}- onto A, for some r. Let wo=tltj (zo). By Lemma 1.7, gaffe D(pro). Denote the set in (2.9) by R(Apzo). Since R(A,,wo) is dense in D(Aflwo). there is hj e R(Aflwo) that is analytic on a neighbourhood of the closure of A, such that e " ff ' “’1' hi "D(A,)< 2(m+l) ‘ By Lemma 1.7, '1 ”ft hj'VJDmf'HUl'Wjw>VJD(n-)="5'Vj'hillD(A,)<—_)'"(212) 18 Since 89,- = TOUT,- and To, I) are analytic curves, v,- extends to be analytic to some neighbourhood G of the closure of 9,- by Lemma 2.5. Choose a compact subset K j of G such that Sz\K,- has two connected components and the closure of 9,- c the interior of K; c K]; Then, by Runge's Theorem, there is a sequence {rn} of rational functions whose poles are off K ,- such that r,I converges to hj - qt,- uniformly on K,- . Hence r,,' converges to (hj- ‘Vj )' uniformly on the closure of 9,- and so r,. converges to hj- V} in M9,). Again r,,(zo) may be assumed to be 0 for all n. Therefore we can choose a rational function r,- whose poles are off K,- such that 8 II hj'Wj -rj "D(Qj)< m . L m+1' Hence, by the triangle inequality and (2.12), II ff - rj "0(0) < 1 After choosing rj for each j, let r = pa + r1 + + r,,, . Then r is in R(9)r\D(9,zo) and II f - r “0(0) < Ilfo-po IID(00)+ ll fl- r1 IID(01)+ + llfm-rm "D(Qm) < 6. Hence R(9)nD(9,zo) is a dense subset of D(9,zo). Q.E.D. Now we are ready to prove our main theorem of this chapter. Theorem 2.13: Suppose 9 is a bounded domain whose boundary consists of m+1 mutually disjoint analytic curves and let 20 e 9. Let (p e M(D(9,zo)). Then 0e(Mtp) = datum). 19 Proof: By Lemma 2.1, we may assume that To = 8U. We first will show that cl((p;a9) c oe(Mq,). It suffices to show that if 0 e cl ((11:39), then M q, is not a Fredholm operator. Suppose 0 e cl((z,.) -> 0. Then there is a j such that z,,-> 1"}. (Use a'subsequence of {2,}, if necessary.) Suppose w,- is a conformal mapping from U onto 11,-. Let a, = we.) Then (tn—2 30 and tp(\|t,(a,,)) = (p(z,,) —> o as n —> ... Let 2.z bethe point evaluation map at z on D(U,0). Then, by (1.15), 1 II2,IID(U) = <2., "z’mm = 2.,(2) = 1;_1og1—— Izl, . Hence || 2. an."D(UO)-’°° as n-)oe. By the Uniform Boundedness Principle, there is a function f e D(U,0) such that supl D(U0)I= Therefore there is a subsequence of {an}, for which we will use the same notation {an} , such that hm = lim f l = oo. 2.14 n:—)co| (ft x (kW) n_)°°| (an) ( ) Let 9* = W}1(9). Let k, be the point evaluation mapping at z e 9* on D( 9*,wo) where wo =u'l.‘(zo). Since flu... -j(wo) e D( 9*,Wo), I flan) - f(wo) I = I < flu. -f(wo), 1513062,“ 5 II fl D(U) ll Ian" D(Q*)- (2.15) 20 By (2.14) and (2.15), ll hill -> oo as n -) co. Define a function f" by lD(9"') k an II lockup“) for each n e N. We claim that f" —) 0 weakly in D( 9*,wo). We must show that D(n*) —> O for all g e D( 9*,wo) as n -> oo. Let g e D( 9*,wo). By Theorem 2.10, there is a sequence [rm] of rational functions in R(9*) n D( 9*,wo) such that rm—> g in D( 9*,wo). Forall ne N and me N, I D(m)l s I D(m)l + I D(m)l S Ilfnlle.) llg-rmllD(Q,) + I D(9,)I. Let e>0. Chooseapositive integer K1 such that lIg-rMIID(n,)< 5'2- if m 2K1. Fix mZKl. Note that l oo, there is a positive integer e |D(m)l < f if nZKz. Hence D(9"') n —> oo. Since we assumed that Mq, is a Fredholm operator on D( 9,20), M ‘P-‘Vj is a Fredholm operator on D( 9*,wo) by Lemma 2.1. Hence there is a compact operator T 21 on D( 9*,wo) such that 1 - M (P-Vj T is compact. Since f,, -) 0 weakly in D( 9*,wo), we have N (1- M‘P-Vj T )(fn) l —) 0. Therefore ID(Q") 1 ' (‘P'Wj )(an) < T(frDsfn >D(Q*) = D(Q*) = < (1 - M(Wj T )0.) , f,, >D(Q*) —) o. (2.16) But, since ((p-Wann) -+0 and | 5 HT" llfnl = IITII, the left D(fl*)| ID(Q*) hand side of (2.16) approaches 1. This contradiction shows that M¢ is not Fredholm. Hence cl(q>;89) c: oe(M¢). To prove the converse inclusion, suppose that 0 e cl((p;89), i.e. (p is bounded away from 0 near 89. Let :1, , 2,, be the distinct zeros of (p in 9. Assume first that zjatzo forall j. Let m(zj) bethe multiplicity ofthezeroof (p at zj. Let E bethe subspace of D(9,zo) consisting of all functions f in D(9,zo) such that f vanishes on { 2,, ,z,) with multiplicity bigger than or equal to tn(z,-) at each 2,. Let fe 5. Then L L - (P e H(9) and q, (20) - 0. To see g is in D(9,zo). observe that (52' =(f'Ip-ftp')/q>2. By the remark (b) following Lemma 1.13, the numerator is square integrable on 9. Since (p is bounded away from 0 near 89, ée mono). 22 Hence f is in the range of 114,, and so E is containedin the range of M9 Note that E=n{Ker 2.2),], : j=1,~--,n and k=0,---, m(zj)-1}. Being an intersection of the kernels of finitely many linear functionals, E has a finite codimension. Since Ker M9 = [0}, Mq, is Fredholm. Now suppose there is j = 1, , n, such that z; = 20, say 21 = 20. Then redefine m(zl) to be (the multiplicity of the zero of (p at 21) + 1, and define E as before. Then, by the same argument, we can conclude that M9 is Fredholm. Thus oe(Mq,) c cl(;89) is true when (p is a multiplier of D(9) on bounded domains 9 in C such that D(9) (2 3(9). By Lemma 2.1 and Theorem 2.13, we have the following corollary. Corollary 2.17: Suppose 9 is a finitely connected bounded domain in C. Let (p e M(D(9)). Then oe(M,,) = cl(tp;a9). CHAPTER 3 CLOSED FINITE CODIMENSIONAL INVARIANT SUBSPACES In this chapter we will study finite codimensional invariant closed subspaces of the Dirichlet space of a finitely connected domain with an analytic boundary. A characterization of those subspaces of the Bergman spaces defined on a large class of bounded domains in C was obtained by Axler and Bourdon in [3]. Also Chan characterized those spaces on D(9) when 9 is a circular domain; see [7]. In his paper, Chan used a Laurent series expansion to prove his characterization, which cannot be applied on noncircular domains. In this chapter, we will establish the same characterization of finite codimensional invariant closed subspaces of D(9) when 9 is a finitely connected domain with an analytic boundary. Recall we assumed that no component of 39 is equal to a point. We start this chapter with the Bergman norm estimation of certain class of functions on U that will be used repeatedly throughout this chapter. r-12. Then Lemma 3.1: For r>1,defineafunction g, on U by g, =_(-:t_I_r—)_ sup { llg, "Ba/)1 r>1} <00. Proof: Let r> 1. Note that ___l_ _°° 11:1... (z-r)2 "Earns-22 where the series converges uniformly and absoluwa on U. 23 Also 1 °° n+1 12 = 2 z" forlz|>l. (3.2) 0-?) n=0 Hence °° n+1 "3,13%: n (r-1)2n§=;o r 2“, +2) by (1.9) =ttr°4 (r-l)2 E n+1 =1tr°4 (r-l)2r—- by (32) "=0 r2" (r2-1)2 ' 115 it -(r+1)2 <4 Q.E.D. The following lemma is well known in general function spaces. For later use, we state it explicitly. Lemma 3.3: Suppose that Hula“ is a norm bounded net in a closed subspace H of D(9). which converges to f pointwise on 9. Then f e H. Proof: By the Banach-Alaoglu Theorem, any closed ball of D(9) is weak“ (hence weak) compact. Therefore {fa} as A has a weak convergent subnet {fat} 66 a , say fan —> g weakly in D(9). Hence, by remark (a) following Lemma 1.5, fap(z) —-) g(z) pointwise on 9 and so f = g and fa.3 -) f weakly in D(9). But the norm topology and the weak topology have the same closed convex sets (see, for example, Rudin [12], Theorem 3.12). Since H is a (norm) closed convex set, f e H. Q.E.D. Remark: A sequence version of the above lemma is still true. We only need to prove that a bounded sequence in D(9) has a weak convergent subsequence. Define an operator T:D(9)—-)3(9) by T(f) = f '. Then T is an isometry. Now, “since 3(9) is separable, so is D(9). Hence any closed ball of D(9) with the weak topology is a metrizable 25 compact set (see, for example, Rudin [12], Theorem 3.16) and so any bounded sequence in D(9) has a weak convergent subsequence. Suppose 9 is a bounded domain such that D(9) c 3(9). If 2 is in 9, then (2 - 2)D(9,zo) is a closed proper subspace of D(9,zo) that is invariant under multiplication by 2. If q is a polynomial that has all of its zeros in 9, then we will see in Proposition 3.12 that qD(9) is a finite codimensional closed subspace of D(9) that is invariant under multiplication by 2. If 2 e C\9_, then (2 - 2. )D(9,zo) = D(9,zo). We will prove in the following two theorems that (z - 2)D(9,zo) is dense in D(9,zo) if 9 is a finitely connected domain with an analytic boundary and 2 e 39. These theorems are key steps toward obtaining a characterization of finite codimensional invariant closed subspaces of D(9) on finitely connected domains with an analytic boundary. By a wedge W), in C, we mean the convex hull of a point 2 (called the vertex of the wedge) and an arc of a circle centered at 2.. We mentioned in Chapter 2 that, if 9 has an analytic boundary, then each boundary point satisfies the external ball condition. Hence, for each 2 e 89, there is a wedge W), in C \ 9 with vertex at 2.. Actually, in order to satisfy this "wedge condition", 39 need only be a Cl boundary by the implicit function theorem and Taylor's Theorem. Theorem 3.4: Let 9 be a simply connected domain with an analytic ' boundary. Then (z - 2 )D(9,zo) is dense in D(9,zo) for every 2 e 89. Proof: Let 2e 39. We know that P(9)nD(9,zo) is dense in D(9,zo) by Theorem 2.10. Hence it suffices to show that P(9)nD(9,zo) c: (z - 2)D(9,zo). the closure of ( z - 2 )D(9,zo) in D(9,zo). 26 Assume that z - 20 e (z - 2)D(9,zo). Then, since (z - 2)D (9,20) is invariant under multiplication by a polynomial, and each p e P(9)nD(9,zo) is of the form q(z - 20) where q e P(9), we would have P(9)nD(9,zo) c (z - 2)D(9,z) as desired. To prove z - 20 e (z - 2)D(9.zo) , we first assume that 2=1e 39. Since an is an analytic curve, there exists a wedge Wl in C \9 with vertex at 1. Assume that there exist as (o, 1) and 8e (0. 45) such that W1={zeCzlz-1|52a,-25Sarg(z-1)525}. Let Gl=lzeC: lz- (1-a)|cos0 lz -1|2 + (r -1)2 - 2lz-ll (r-1)c055 27 (lz-ll-l-(r-1))2 Max { lz-llzsin25'(r-1)zsin28 } . (3.6) If lz- 1|2r- l,then the numeratorof the right hand side of (3.6) is less thanorequalto 4lz-112. When r-12lz-1l,thenumeratoroftherighthandsideof (3.6) islessthan orequalto 4(r-1)2. Inanycases,wehave the right hand side of (3.6) s ,4, . (3.7) sm 8 Hence there is a constant K such that H g, "D(Gz) = H gr. "3(62) 5417+ ll(r2- r)/(z- r)2||3(62) +llzo(r-1)/(z-r)2l|B(Gz) by (3.5) S‘f1?.*-sir‘:28["(rz-r)/(z.r)2"B(GI) +l|zo(r-1)/(z-r)2IIB(Gl)] by change-of-variables and (3.7) < K. by Lemma (3.1) Therefore sup{llg, "D(Gz) : re (1,1+a)] (z - zo) pointwise, and (z - 1)D(9,zo) is a closed subspace of D(9,zo), z - 20 is in (z - 1)D(9,zo) by Lemma 3.3. For general 2e 89, suppose there exist toe C, ae (0,1), and 8e (mg) such that W;={zeC: Iz-lls 2a, arg(to-2)-26 S arg(z-1)S arg(to-2)+28}. Let L={te C: lt-1| 1. Hence, in order to prove that h is in (z - l)D (9,20) , by Lemma 3.3, it suffices to show that sup { ||(z- 1)z—f’—rIID(Q):re (1, 1 +a) I = w —_—-Cj ((04,), - (zo-w)*"((,-;,.)k - (zo_§j)k)] Then , by induction hypothesis, gj is in (z - 1)D(9,zo) for all j. Note that sup{llgj'll” :je N} 0. Hence sup [ llgjlle): je N ] 0 and let sup{lz-2II: ze 9 } =K. Let f e D(9) = (z - 2.1)D(9) . Then there is a function g e D(9) such that 35 II (2 - 2.1)g - f "D(9) <%. For g e D(9) = (z - 22)D(9) ,there is a sequence of functions [ gn } in D(9) such that ll (2 - 22)g,, - g "D(Q) —) 0 as n -9 co. Since the inclusion map from D(9) into 3(9) is bounded, we have ll (2 - 22)g,, - g "3(a) -) 0 as n —) co. Hence there is a function go in D(9) such that II (2 - 22)g0- g "0(a) <33]? and II (2 - 22)g0 - g "3(a) < g . Therefore II (2 - 2.1)(2 - 22)g0 -f||D(g) 5 N (Z - 3«1) [(2 - 2380' 8] "0(0) + ” (Z ' A4).? -f "D(9) 5 N (Z - 2.2)g0 - g "3(9) + K ll (2 - 2.9,)go- g "Dan-1' ll (2 - 2.1)g -f "D(Q) < 8. Thus f is in (z - 2.1)(2 - 2.2)D(9) and so we proved (3.16). Since k has only finitely many zeros, we can conclude that W9) = D(9) by repeating a similar argument. Suppose f isin qD(9)=q (Eb-(TB). Then f = qg for some g inD(9). For g, there is a sequence of functions [gnl in D(9) such that ll k g" - g "D(9)-’0 as n -—> 0. Hence " 4k gn'fllD(9) = " 4" 8n ' 48 "D(9) = " q(k 8n ' 8 ) "D(9) S ll q' (k gn- g ) + q (k gn - g )' "3(9). (3.17) Since q and q' are bounded on 9, and the inclusion map from D(9) into 3(9) is bounded, the right hand side of (3.17) approaches 0 as n -) 0. Hence q(kD(9) ) c qu(9) . Therefore 36 qDIm= (NW) C m= 1272223") c E. (3.18) Hence dim (D(9) 140(9) ) =degree ofq by Proposition 3.12 Sdegreeofh Sdim(D(Q)/E) bythechoiceofh s dim (D(9) [qD(Q) ). by (3.18) Hence dim (D(9) /qD(Q)) = dim (D(Q) IE) and so, by (3.18), E = qD(9). Q.E.D. Cor 3.19: Let E be a finite codimensional closed subspace of D(9) where Q is a finitely connected bounded domain with an analytic boundary. Then the following are equivalent. (1) 2E c E (2) tpE c E for all (p e M(D(9)) (3) E = qD(Q) where q is a polynomial with all of its zeros in 9. Proof: (1) implies (3) by Theorem 3.15. (3) implies (2) since (pE=