ItWWWNWHUMWWW“WM 128 158 __THS IC em STATE umvensmr LIBRARIES “Willi lilillilll’lll ill i Nil 3 1293 01411 4643 \l ‘ ll T143813 This is to certify that the dissertation entitled Essentially normal multiplication operators on the Dirichlet space presented by Jaroslaw Lech has been accepted towards fulfillment of the requirements for Doctor of Philosophy 'Mathematics degree in $4.,ka Major professor Date 16- M [6195’ MS U i: an Affirmative Action/Equal Opportunity Institution 0-12771 LIBRARY M'Chigan state Unlverslty PLACE DI RETURN BOX to remove“: checkout from your record. TO AVOID FINES Mum on or More data duo. DATE DUE DATE DUE DATE DUE gyzscw } MSU IsAn Affirmative Adlai/Ema! Oppommlty Institution Wan-so.- ESSENTIALLY NORMAL MULTIPLICATION OPERATORS ON THE DIRICHLET SPACE By J aroslaw Lech A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1995 ABSTRACT ESSENTIALLY NORMAL MULTIPLICATION OPERATORS ON THE DIRICHLET SPACE By J aroslaw Lech This thesis deals with essential normality of the multiplication operators on the Dirich- let space D. We construct an example of a multiplier of the Dirichlet space with non-compact self-commutator and show that the essentially normal multipliers of D can be characterized in terms of a certain capacity condition. Similar problems were studied for the Bergman and Hardy spaces where they can be reduced to the question about compactness of Hankel operators. ACKNOWLEDGMENTS I would like to express my sincere gratitude to Professor Sheldon Axler for his help, advice, and encouragement. I also would like to thank Professors Wade Ramey and Alexander Volberg for their lectures. iii Contents Introduction 1 l Compactness of the self-commutators in the Bergman and Hardy spaces 2 2 Multipliers with compact self-commutators 4 3 Multipliers with noncompact self-commutators 9 4 Questions and comments 18 Bibliography 21 iv INTRODUCTION Let U be the open unit disk in the complex plane C. The Dirichlet space D is the Hilbert space of analytic functions f (z) = 23,20 anz" on U such that Izié = 2 nlanlz < 00, 11—] m) = o and llflli) = [U |f’(2) where dA denotes the usual area measure. An analytic function ap on U is called a multiplier of D if 900 C D. The set of all multipliers of D will be denoted by M (D) Each multiplier generates a bounded multiplication operator M¢ on D defined by Mg, f = 90 f for f 6 D. It is easy to see that non—constant multipliers of D are never normal, which raises the question: How far are the multiplications on D from being normal? In this thesis we investigate the essential normality of the multipliers of D. A Hilbert space operator T is essentially normal if the self-commutator T‘T — T T‘ is compact. Nice characterizations of all bounded analytic functions that generate essentially normal multiplications have been found for the Bergman and Hardy spaces. We recall those results in Chapter 1. In particular it is known that every multiplication oper— ator by a function in D + C on the Bergman and Hardy spaces does have compact self-commutator. It turns out that despite the fact that M (D) g H °°(U ) n (D + C) multipliers of D with non-compact self-commutator exist. Chapter 3 contains a con— struction of such a function, answering negatively the question raised in [AS] by Axler and Shields. In Chapter 2 we convert the problem of compactnes of Mng — M¢M;, to a question about the compactnes of the operator Mw: D —> B of multiplication by cp’. We also give a characterization of essentially normal multipliers of D in terms of a certain capacity condition. The thesis ends with a discusion of related questions in Chapter 4. Chapter 1 Compactness of the self-commutators in the Bergman and Hardy spaces Recall that the Bergman space B is the Hilbert space of all analytic functions f (z) = :20 anz" on U such that 00 dA ufn'g = [(111127 = z: IanIQ/(n +1) < oo. n=0 and that the Hardy H 2 space is the Hilbert space of all analytic functions f (z) = 2310 anz" on U such that 00 ‘21r _ d6 2 _ :9 2 = 2 llfllm — (gig/0 |f(re )I —27r Eilanl < 00. 72:0 It is well known that for the Bergman and Hardy spaces the set of all multipliers is equal to H °°(U ) (the set of all bounded analytic functions on U). We will use M: and Mg2 for the multiplications on B and H 2 respectively. Let us consider the orthogonal projections: 13,.3 : mad?) —+ B, and d0 P112 1L2(0U,-;r-) -—) H2, where the functions in H 2 are identified with their boundary values. For any <15 6 L°°(U, 9%) the Bergman space Hankel operator Hf : B —> B‘L is defined by: Hff =(1 - PB)(¢f) where B‘L is the orthogonal complement of B in L2(U, if). 2 3 For any 45 E L°°(6U, %) the Hardy space Hankel operator Hf? : H2 —-> H3“L is defined by: 2 Hf f = (I - mew) where H u is the orthogonal complement of H 2 in L2(0U, $9). The following Proposition reduces the problem of the essential normality of Mg and M5,: to the one about the compactness of Hankel operators. The proof can be found in [A1] (Proposition 3). Proposition Let (p e H°°(U). Then B: B B B. B' B M¢ M“, —M¢Mw =H¢ Hg, and 2... 2 2 2,, H7. H2 Mf Mj,’ —Mfo :11, HP. The next two theorems characterize the compactness of Hg and Hg), and hence the essential normality of Mg and Mfg. Theorem A Let «,9 E H°°(U). Then Hg is compact if and only if |¢'(Z)l(1-|Z|2)-+ 0, as |Z| —* 1 Theorem B Let (,9 E H°°(U). Then Hg? is compact if and only if 1 d9 m/Il99-991|§;-+0,85 III—*0, where I denotes subarcs ofaUand 991 denotes the average ofgo over 1. Theorem A is due to Axler and can be found in [A1] (Theorem 7). Theorem B can be found in [Z] (Theorem 9.3.2). Analytic functions that satisfy the condition in Theorem A are said to be in the little Bloch space 50. Analytic functions satisfying the condition in Theorem B are said to be of vanishing mean oscillation (VMOA). It is well known that D + C C ,30 fl VMOA, yet as we will see it is possible to construct a function in M (D) with non-compact self-commutator. Chapter 2 Multipliers with compact self-commutators In Chapter 1 we have seen that the self-commutators of multiplication operators on the Bergman and Hardy spaces are products of a certain Hankel operator and its adjoint. This can not be true in the Dirichlet space because the self-commutator is not positive even for elementary multipliers such as functions analytic in U. The main goal of this chapter is to find the right substitute for the Hankel operator in the Dirichlet space i.e., to find an operator whose compactnes is equivalent to the essential normality of Mg, but that is easier to deal with than the self-commutator. A few more definitions are in order. The harmonic Dirichlet space D), is the Hilbert space of functions f on the unit circle T for which llflli), = |f(0)|2 + Z |n||f(n)|2 < 00, fl=-OO A where ( f (n)) is the sequence of Fourier coefficients of f. It can be shown that . CM 2 _ 2 2— Ilfllo, — mo» + [U IVPlfll 1r « a 2. f(e“’)—f(e“) = 0 2 m )I + [0 [0 eta _ eta where P[ f] denotes the Poisson integral of f. (The first of these equalities follows from an easy computation; for the proof of the second one see [D], pp. 307—311.) Since each function in D can be identified with its boundary values, we may think of D as being a closed subspace of Dh. This allows us to consider the projection map PD: D}, —» D. We begin by showing that if (,0 is a multiplier of D, then 95 multiplies D into the 2 d0 d5 27??? harmonic Dirichlet space. As usual Hap”.>0 4g supzeu |cp(z)|. Lemma 1 If<,o E M(D) and f E D, then 95f E Dh. 4 5 Proof. By assumption cp E M(D ) so ”90”.» < 00. We have IWW f(6 '9)- cp(6“)f(6")|’ 2|(6‘")f(6 '9)- (6'——"_)f(6 ")|2+2|s0(6"’)f f(6 ")- D by T; f = PD(¢ f ) where P is the projection map from D), to D. Lemma 2 Let cp E M(D). (a) The operator T, is unitarily equivalent to the adjoint of multiplication by 99 on B. (b) M; — T; = M;,R where Mw: D —-> B is multiplication by 90’ and R: D —> B is a unitary operator. Proof. (a) Let en(z) = W, 65(2) 2 fizn‘l for z E U,n = 1,2... It is easy to check that (e,,);,'°=l form an orthonormal basis in D and that (ef),i,°°=l form an orthonormal basis in B. Let R be the unitary operator from D to B which takes f to f’, and let ( , ) D, ( , )3 be the inner products in D and B respectively. Finally let (,0 = {2:0 anz" E M(D). Direct computation shows that DI m n—m -f < (T¢en,em)p = (PD¢en,em)D 2 J77 m l m _ n (2.1) 0 if m > n and mom, 'f n < n 1 1 (R.Mf.RememlD = (Rem‘PRemlB = (65,996ng = fifi _ 0 if m > n Hence T; = R'ME‘R. (2.2) (b) Using part (a) gives ((MS - T¢)f,g)n = (Leah) - (f’,9eg'>3 = (f’,99’9)3 = (MSIRf,g)D as desired. D We need the following lemma, whose proof can be found in [AS] (Theorem 9). Lemma 3 Let cp be a holomorphic function on U and let MW, be the operator of multiplication by (p, (a) If MW: D —+ B is bounded then , l SUP I99 (Z)l(108 —2)1/2(1-|Z|2) 5 ”MW“ Izl<1 1 — '2] (b) If le: D —-» B is compact then , 1 |¢(z)l(10g1_|zl,)"2(l-lz|2)-+ 0 as|z|->1 The proof of Theorem 1 uses the following form of the Fuglege’s theorem: Let a,b,c be elements of some C‘-algebra. If a and b are normal and ac = cb then a‘c = cb“. (For a reference see [R], Theorems 12.16 and 12.41.) Now we are ready to prove the main result of this section. Theorem 1 Let 9p 6 M(D). Then M,p is essentially normal if and only if MW: D —-> B is compact. Proof. Since (,9 E M(D), Mgo’: D -) B is bounded and by Lemma 3(a) |¢'(z)|(1-|Z|2) -* 0 as |Z|-*1~ This, as was shown by Axler ([A1], Proposition 3 and Theorem 7) implies that the operator N,p of multiplication by (,0 on B is essentially normal. 4: By Lemma 2(b) M,p — T5 = R'Mw, so our assumption, Lemma 2(a), and the remark made above imply that M“, is a compact perturbation of an essentially normal operator. => Denote K = R‘Mvt where R is the unitary operator taking f to f’. By Lemma 2(b) M; - T; = K" (2.3) and Mw — T; = K. (2.4) We clearly have M? M. = M..M. and thus R-leRR-IM. = R"M..M.. (2.5) Since R‘1 = R“, (2.2) and (2.5) imply that T‘sK = KM.. By assumption, Lemma 2(a), and the remark made at the beginning of the proof both T. and M. are normal in the Calkin algebra. Thus using the Fuglege’s theorem T.K 2: KM; and K’T‘; = M.K" in the Calkin algebra. (2.6) Equations (2.3), (2.4), (2.6) imply that in the Calkin algebra o = MgM. — M.M;, = (T. + 1mm; + K) .— (T5, + K)(T. + K“) = (T.T;; — TgT.) +(T.K — KT.) + (K'T; — T51?) + (K‘K — KK“) ‘—-—~——’ 0 = K (M; — T.) + (M. — T5) 11" + (K‘K — 1m") hw—’ hr—J K- K = KK" + K "K. Since both K K "' and K ‘K are positive, they have to be 0 in the Calkin algebra, and hence K is compact, which forces M. to be compact. Cl Remark. In [BDF] Brown, Douglas and Fillmore studied essentially normal operators. One of their results was the following: If S is essentially normal and ind(S — AI) 5 0 for all A outside the essential spectrum of S, then S is unitarily equivalent to a compact perturbation of a subnormal operator. Here ind denotes the Fredholm index. Notice that M.—AI = M.-. has trivial kernel (ifgo is non-constant) so ind(M._,\1) S 0 for all A not in the essential spectrum of M.. Theorem 1 says that if M. is essen- tially normal then M. is compact, and since M. = R“ MfR+ R“ M.I, M. is unitarily equivalent to a compact perturbation of the multiplication on B—one of the main examples of subnormal operators. Thus Theorem 1 gives an explicit example of the phenomena discovered by Brown, Douglas, and Fillmore. In [S] (Theorems 1.1 and 2.3 ) Stegenga found a description of all analytic (,9 such that cp’D C B in terms of boundary behavior of (p. His result says that M.:: D ——) B is bounded if and only if [Um lso’lsz = 0(Cap(U I.» where (11') is any finite collection of disjoint subarcs on the circle, 5(1) denotes the “square” in the disc with side I , and Cap denotes the logarithmic capacity. 8 In [RW] (Corollary 3.1) Rochberg and Wu proved that compactness of M.: is equivalent to a “little 0” version of the Stegenga condition. This and Theorem 1 gives: Corollary 1 Let (,0 E M(D). The operator M. is essentially normal if and only if ’ 2dA = 0 Ca I- [Usun to I < p(U .)) where (15) is a finite collection of disjoint subarcs on the circle, and 5(1) is the “square” in the disc with side I. Chapter 3 Multipliers with noncompact self-commutators In this section we will show that there are multipliers of D for which the corresponding multiplication operator is not essentially normal. By Theoreml it is enough to construct cp E M (D) such that the operator M.:: D —-> B of multiplication by cp’ is not compact. We will do this in two steps. Theorem 2 below shows the existence of a function cp holomorphic on U, with M.:: D —+ B bounded and , I [90(Z)l(1081—_'_—|z—|§)1/2(1—|Z|2)7"’ 0 as [Z] —’ 1- For such a (p the operator M.: is not compact (see Lemma 3). A method of making (,0 bounded without losing any of its properties is given by Corollary 2. As a result we will get a multiplier (p with noncompact M.:. The extended Dirichlet space D is the Hilbert space of all analytic functions f(z) = 23:0 anz" on U such that I 2dA fume» 7 < oo. The norm on D is defined by 2" . d6 dA °° ”in = [0 |f(6'9)|2§;+ Ulf’(z)|27 = Z(n+1)la.I’-’. n=0 where d0 denotes the usual Lebesgue measure and f (em) is the nontangential limit of f (d0 almost everywhere). It is clear that D and D differ only by one dimension and that the norm [I ”D restricted to D is equivalent to [I “D. Thus the operator M.: from D to B is compact (bounded) if and only if it is compact (bounded) as an operator from D to B. For technical reasons the next theorem uses D instead of D. 10 Theorem 2 Let 0 < c < 1. There exists (,9 analytic in U and a sequence (2,.) C U converging to 1 such that I. 1 1— I2 (2W1 — 12.42) —> c as n -+ co lse'(zn)|(log 2. ||M.,||p_.3 S 1, where [I |]p_.3 denotes the norm of M., as a multiplication by cp’ from D to B. We will need a few more lemmas before proving Theorem 2. We will adopt the following notation: kw(z) = _i log 1 and Kw(z) = 1 wz —1122 (1 —wz)2' It is easy to check that f(w) = (f,lcw)p for all f E D and f(w) = (f, Kw)3 for all f E B. The functions kw and K. are called the reproducing kernels for D and B respec- tively. It is not hard to see that for any finite set of distinct points w1,w2, . . .wn in U the corresponding families (1:...) and (K...) are linearly independent and that the norms of Is. and K. are llkwllv = (kw(W))% = | ((1 g1_|w|2)% and ||lela = (deDi = 1——1Tu3|_2 Notice that if , “K... Ila lee (Zn)[ = cm for some sequence (2,.) C U converging to 1 then condition 1 of the theorem is clearly satisfied. Moreover if M.:: D —i B is bounded then (Millimflv = (KwHP'fla = (W’fales = 99’(W)f(w) = 99’(w)(fakw)v = (90’(w)kwv fl‘D for all f E D, and hence M;,Kw = ’(w)kw. 11 This suggests that we may specify the values of cp’ using the operator M5,. More precisely we will construct a sequence (2..) C U converging to 1 and an operator Ac : B —+ D with ”AC“ 31 and —'C”I‘ Zn “8 Ciikzn ”D in such a way that Ac = M5,, for some cp. This will give us a function cp and a sequence (2..) with all the required properties. The idea described above and many techniques used in the proof of Theorem 2 come from the preprint of Marshall and Sundberg (see [MS]). First we will prove: ACK 2.. _ k." for n =1,2,3,... Lemma 4 Let 0 < c < 1. Then there exists a sequence (2,.) C U such that 2,. -—> 1 and the operators Afi: span(K.1 .. . K...) —> span(k., . . . k2") defined by CIIKz. llB AC "K...— _ ”'6. Ho kz.for 2,...,n. satisfy ”AS,” S 1 for all n. Proof. Notice that if the families (fl)?=1 C B and (g.~)f‘=1 C D are linearly independent and L:span(f1,f2, ° ' 'fn) —) Spa’n(glag2a ' ° gn) is defined by L f.- = agg. then ”L“ S 1 ”M: bit-HI?) S H Zbifillze for an (50:21 C C i=1 5:] (Z b.-f.-,Zb.-f.-) )3 - (2b.- a.g.-,Zb.- a.g.-)3 2 0 for all (b.)?__., C C i=1 i1: i=1 it Q 2:“ ((fiafle “aiajigivgjlv )bibj >Of0r all (biliz 1CC i=1 j=l Hence ”L” S 1 4:) {(f.,f_,~)3 — a.c‘zj(g.-,gj)3}.-,j=1,2,,_,,,. is positive semidefinite, (3.1) and all we have to do is to find a sequence (2,.) C U such that 2.. —+ 1 and the matrices , , [le IIB lleIIB (I‘ZHI‘Z )3 — C2 i J (kznkz )D { J ||k.,||p ”1‘22”” 1 i.j=1,2,...,n (1(2‘31{2')B (kZ'akZ )9 = IIKz.|IB||Kz.IIB( ' ’ _.2 ' ’ i IIKz.I|BIIKz.IIB nunvumlv 12 are positive semidefinite for all n = 1,2, . . . . Since the matrix (M. IIBIIKz, llali.j=1.2.....n is a Gramian (hence positive semidefinite), and by Schur’s lemma ([HJ], Theorem 7.5.3) the entry by entry product of positive semidefinite matrices is positive semidef- inite, it will be enough to construct a sequence (2,.) C U such that 2,. --> 1 and the matrices K K k k { < z.) 2,)3 _ 62 ( zn 2,)D } (3.2) llK..|lB||K.,||B Ilkz.||v||kz,||v .-,,-=.,.,,,,,.. are positive semidefinite for all n = 1,2,3, . .. . We will define inductively a sequence (2,.) for which 1 — 1/ n < 2,. < 1 and (Kr, Kz>B (k2, k: )D } det{ ' ’ — c2 ' ’ > 0 (3.3) IIKz.|IBIIKz.IIB Ilkz.llvllkz. no ,,,,, ,. for all n. This implies that the matrices of type (3.2) are positive semidefinite for all n by standard linear algebra ([HJ], Theorem 7.2.5). For n = 1 let 21 be any real number between 0 and 1. Then 1-by-1 matrix of type (3.2) consists of the single entry 1 — c2 and (3.3) is clearly satisfied. Suppose we constructed 2., . . . , 2N_1 such that 1 — 1/i < 2.- < 1 and the condition (3.3) is satisfied for each n = 1,2,. . . ,N — 1. For any real 2N we can expand by minors along the last column, getting K2) 2 krikz det{ < t [(1)8 -C2 ( I I)” } Ile.||BIIKz,I|B llkz.|Iv||kz,IIv .-.~=1,2,...,~ = (1 — c2)det{ 1(ZWKZ' kz'akz‘ < . J)B —C2 < a J>D } +A lle.|IB|IKz,IIB llkz.||v||kz,||v .,,-=.,.,,,.,N_. where A is the sum of terms each of which contains a factor (KanZNlB __ c2 (kzaikZva ||K2.I|BI|Kz~|IB Ilkallvllkmllv _ (1- 23x1 - a) 2 log _ (1 - MN)? (10g flit/200g 7:17)”? i N for some i = 1,2, . . . , N — 1. Each of those factors can be made as small as we want by making 2N sufficiently close to 1, so there is a 2N such that 1 — 1/ N < 2N < 1 and Kz-,Kz~ k2" Z IAI<(l—cz)det{ < ' ’>8 —c2 ( 'k’)v } . IIKzJIBIleJIB Ilkz.llvllkz,llv .-,,-=.,2,,,,,N_1 This implies (3.3) for n = N. D The following lemma helps us to extend operators Af, and will play a crucial role in the proof of Theorem 2. 13 Lemma 5 Let 2., 22, . . . , 2,. be any sequence of complex numbers in U. Suppose the operator 5' : span(K.,,Kz,, . . . , K2”) —» span(lc,,, k2,, . . . , k2") defined by 5K2. = mic... fori = 1,2,.. . ,n and some collection of complex numbers r1,r2, . . . ,rn satiSfieS ”5” S 1. Then for each 2 E U there exists a complex number 1' for which the operator 5.. : span(K.,,K.2, . . . ,Kzn,Kz) —i span(kz,,kz,, . . . ,k k.) Zn, defined by .3er.. = ink... for i = 1,... ,n and 5'er = rk. satisfies “S,“ S 1. Proof. Fix 2 E U. The map t —+ “S.” is continuous on C and goes to 00 as |t| —+ 00. Thus there exists r E C such that IISrII = gggIISill- Denote by Hf the subspace of span(K.1 , K22 , . . . , K2", K.) orthogonal to K, and by Hf the subspace of span(kz, , k2,, . . . , k2”, k.) orthogonal to k.. Let P3 : span(K.,,Kz,, . . . , KW K.) —. HE P” :span(kz,,k.,,.. .,kz,,,kz) -—+ H? be the orthogonal projections , 1?... = PBKZ. for i = 1,2, . . . , n and lit... = P” k... for i = 1,2,. . . ,n. It is easy to see that ., (1(z-aKle I z- = [(2 — ——'—__—'I{z ‘ ' ' 0 for all n = 0,1,2,.... Clearly a0 = 1, so let’s assume that a.- are nonnegative for i = 0,1,. . . , n. Write 15 1 1 °° 1 210g1—2 zgcnz" wherecnz—n‘k1 Wehave 00 1 (Zanzn) (2c,2’)- — “_z)2=1+22+322+423+... hence aocn+alcn_1+...+a,.co=n+l and ago.“ + alcn + . . . + an+1co = n + 2. Multiplying the first line by (n + 2) / (n + 1) and subtracting from the second gives n+2 n+2 ”cu-['1 n+lcn—1_Cn)+---+an(n+l Because a. is positive for i = 0,1,2... ,n and (c,.) is decreasing, we have a...” > 0, as desired. The rest of the proof is devoted showing that amt-160: ao(n _Cn+1)+ al( 61 — Co)- {woowij - wiowj0}i,j=l,2...,n is positive semidefinite. We have woowi j - 10:0ij = w00(w00 + 112.5 — wgo — 1510) — (woo — wi0)(w00 _ wjo) = woo z: “(2on + 25‘ f+ 2*23— 2:23) — (woo - w£0)(w00 — 1510) = 1000 Z ak( 2f — ZO)( )(Zk — ZS) — (woo — 10.0)(1000 — wig). Thus for any complex numbers b1, b2, . . . , bn n 2 Z bgbflwoowij — wgo’LDjo) i=1j=1 = meakZZb.b( ,- (2]c —20) )]-‘(2 -23) k=0 i=lj= l " Z Z b.b,-(w00 - wioxwoo - 11130) i=1j= 1 = wm2ak|2b.( 2k —20)| 2-[Zb.( (woo- w.-o)|2 i= i=1 n m “’00 Z “kl 2: Mi? - 55W 421%: “1:259? - 5§)|2 k=0 {:1 t-l k-O $1 $2 16 Notice that since an > 0 for all n = 0,1,2,. . . , both 51 and 52 are nonnegative and by Hblder’s inequality 1 52 = 12(aézo) azzm -23» i=1 _<_ gaklzoIZkZakIZb( 2f —20)| i=1 = 1120()E:ak|z:b-(2:c —20)| ==.51 i=1 Hence 51 — 5'2 2 0, as needed. D We now prove Theorem 2. Proof of Theorem 2. Fix 0 6 (0,1). Let (2“) be a sequence of complex numbers promised by Lemma 4. Fix n and consider the operator AS, defined as in Lemma 4. Let {2; +1, 2;”, 2;”, . . .} be any countable dense set in the disk. Lemma 5 allows us to extend A; to an operator Lf, : span(K,,, .., K2", K4”! , K25“, . . .) --> span(kz1 , .., k2", [941“, k211+2"") in .such a way that LiKz' = c”———Kz'“8kz, for 2: 1,2,3. ..n, llkz.||v LZKZ: zfz:kz: fori=n+1,n+2,n+3..., and “Li. H <1 K2: Since span(K,1 , K2, ,. .,Kzn, K21“ , n+2, . . .) lS dense in B, LS, extends by continu- ity to a bounded operator from B to D. Moreover for each 2 E U Lng = i‘zkz with r2, = c(||Kz,||B/||kz, ||p)kz, for i = 1,2,3, . . .. Define 1b,,(2) = r,. Then (14:7le = (Li‘fv1{z)3 = (famkzh) = ¢n(z)f(z) for each f E D and 2 E U. Thus L3: is a multiplication by «bu. In particular 11),, is analytic and if cpn denotes any antiderivative of #2,, then L": M ‘ .The norms of L” are uniformly bounded by 1 so there 13 a subsequence of (Lff) that converges weak' to some operator L". Clearly there exists cp analytic in U with L°'_ — Alp. Thus C ”1(21‘ ”3 llkz. Ilv Now we can answer the question discussed in the introduction: ||M¢1||D_.3 S l and |ap'(2n)| = for i = 1,2,3,....CJ Corollary 2 There exists a function 1b 6 M(D) such that Md, is not essentially normal. 17 Proof. Let c 6 (0,1) and 9p be the function constructed in Theorem 2. Then Mw: D -—> B is bounded, wan (log firm-12.1214 c as n —> oo. for some sequence (2,.) C U converging to 1, and cp’(2,,) -> 00. Let K be compact set of positive area measure contained in the complement of 0. Let 1b = g 0 9p. Then 4) is bounded, MW is bounded, and , l W (an1 (log WWU — lznlzl 7" 0 as lznl —" 1- Thus 1b E M (D) and MW is not compact.Cl In [AS] (Corollary 4) Axler and Shields showed that M (D) is nonseparable in the operator norm. Let W be the space of all holomorphic functions (p in U such that Mg”: D —-+ B is bounded with the operator norm. It is no surprise that W also turns out to be nonseparable. Corollary 3 The space W is nonseparable. Proof. Fix c E (0, 1). A minor modification of Lemma 4 and the proof of Theorem 2 lead to a sequence (2”) in the unit disc with [2”] > 1/2 and 2n —> 1 such that for any sequence (an) consisting of 1’s and -—1’s there exists a function 90 E W satisfying “KznllB ”kznllv Let (an) and (bu) be any two different sequences of 1’s and —1’s, and 1,9, 11) be the corresponding functions in W with I Kzn I 99(271) = can'lili'E—llllf and 1b (Zn) = Cbn 90%an = can lle..||B ”anIID Then by Lemma 3 , 1 ”MW — MW“ 2 Slip '90 (Zn) _ ¢’(2n)l(log '1—__|z—|2)1/2(1_ lznlz) > sup clznllan — bnl Z c n Since the set of all sequences of 1’s and —1’s is uncountable, W has to be nonseparable. D Chapter 4 Questions and comments Let H be a separable Hilbert space, and let (c.0311 be an orthonormal basis for H. A bounded operator A on H is said to be in the Schatten class 5,, if and only if f((A'AV/zen, en) < 00. 71:1 Theorem 1 shows the equivalence between the compactness of the operators M;M¢— MwM; and MW. It would be interesting to know whether there is any connection between the Schatten classes of M;M,, - MwM; and MW. Question 1 Is Mng, — MwM‘; E 5,, if and only if MW 6 52p? The technique used in the proof of Theorem 1 can not be applied here because F uglede’s theorem modulo Shatten p-classes is not true. Operators belonging to the class 5'1 are often called trace class. For a trace class or positive operator A the sum zf=1(Aen,en) is independent of the choice of the orthonormal basis and is called the trace of A. It is well known that for 90 E H °°(U ) tr(1wf*Mf — 1145114?) 2 Noll?» and 2 ,, 2 2 2,, MM: Mi! ‘1”: M: l=||99||b1 where tr denotes the trace. The left-hand sides in both of the formulas above always make sense because the self-commutators in H 2 and B are positive. It is easy to see that for polynomials the same formula holds in the Dirichlet space. If we assume that M;M,, — M,,M; e 31 then tr(M;M,, — M,M;) = tr(RM;,M,, — MwMgR") = /U((RM;M,, — M,M;,R‘)K., [(2)8 dA(z) 18 19 = /U((Mf-Mf _ Mfot)K,,I{z)B C01(2) v2 llvllp + 2Re/U((Mf‘M,,.R* — M,.R'Mf')1<.,Kz)B dA(2) c 1', a + /U((RM;.M¢,R’ — M¢.M;,)K,,K,)B dA(z). i; The second equality follows from the fact that for a positive or trace class operator TonB trT = /U (T1{,,Kz)BdA(z) (For a reference see [AF P] Proposition 3.5). Using reproducing properties and Fubini’s theorem one can show that 11 = 12 = 0 provided that 1 I2 /U [cpl log 1 _ lzlsz(2) < 00 Hence any function go in M(D) satisfying1 [Iv’l—l—I2log —112 dAz<>0 < ||M¢||. Since the spectral radius of M9,, is equal to ||cp||oo and for hyponormal operators spectral radius coincides with the operator norm, no such function can generate hyponormal multiplication operator. The inequality turns out to be false in general but the following question remains valid Question 2 Is every non-constant multiplication operator on the Dirichlet space not hyponormal? Note that since 00 ((Mng — McpMilelaellD = “99% + Z |a,,|2, 71:] no non-constant function that satisfies (4.1) can generate a hyponormal multiplication operator. A positive answer to Question 2 would stand in sharp contrast with the 20 Bergman and Hardy spaces where every multiplication is subnormal. BIBLIOGRAPHY Bibliography [Ag] J. Agler, Interpolation, preprint, 1986. [AFP] J. Arazy, S. D. Fisher, and J. Peetre, Hankel operators on weighted Bergman spaces, American Journal of Mathematics, 110 (1988), 989-1054. [A1] S. Axler, The Bergman space, the Bloch space and comutators of multiplication operators, Duke Math. J. 53 (1986), 315—332. [A2] S. Axler, Interpolation by multipliers of the Dirichlet space, Quart. J. Math. Oxford(2), 43 (1992), 409—419. [AS] S. Axler, A. L. 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Rochberg, Z. Wu, Toeplitz operators on Dirichlet spaces, Integr. Equat. Oper. Th. 15 (1992), 325—342. [R] W. Rudin, Functional Analysis, 2nd ed., McGraw-Hill, New York, 1989. 21 22 [Sar] D. Sarason, Function Theory on the Unit Circle, Virginia Polytechnic and State University, Blacksburg, 1978. [S] D. Stegenga, Multipliers of the Dirichlet space, Illinois J. Math., 24 (1980), 113—139. [T] G. D. Taylor, Multipliers on D0,, Trans. Amer. Math. Soc. 123 (1966), 229—240. [Z] K. Zhu, Operator Theory in Function Spaces, Pure and Applied Math., no 139, Marcel Dekker, New York, 1990. MICHIGRN STATE UNIV. LIBRARIES [[11111][I[I][ll[|[[[|[[|[[[[|[|[[[1] 31293014114643