Integral Equations and Operator Theory Vol. 14 (1991)

0378-620X/91/020213-1651.50+0.20/0 (c) 1991Birkh~user Verlag, Basel

D U A L I T Y A N D M U L T I P L I C A T I O N O P E R A T O R S 1

Pratibha G. Ghatage and Shunhua Sun

We describe a space of functions contained in/:~ f3 C(D t3 G) but not necessarily i n / / . We give a representation of these functions as bounded multiplication operators on the Bergman space/:2 and identify the subspace consisting of functions which induce compact multiplication operators.. We also describe a new C*-subalgebra of/:~176 which we conjecture to be a proper super-set of L/.

Introduction. In our first paper we investigated a space of functions which is

defined in terms of a logarithmic growth condition near the boundary. Most of the well-

known results deal with growth rate of the form (1 - [z[) a (a # 0) as Izl --* 1. However, if

one looks at the Bloch space as the dual of Bergman space/3~ then the growth condition

that the Bloch functions satisfy involves log(1 - [z}) -1. Hence we used this rather natural

condition to define a space of analytic functions on the disk whose growth rate near the

boundary is dominated by log(1 - [z[ ) -1 . This space X, which sits on the edge of the Bloch

space is in fact part of a triple, (X0, ]I, X) satisfying X~ = Y and Y* = X, where Y is a

subspace of/3~ consisting of discrete sums involving the normalized Bergman reproducing

kernel indexed by points of the disk.

1 Most of this research was done while the second author was visiting Cleveland State University. He would like to thank the Mathematics Department for its hospitality. He would also like to thank the NNSFC for partial support.

214 Ghatage and Sun

In the first section of this paper we describe the same space X as part of another

triple (X0, Z, X) where Z is a subspace of the Hardy space H I defined analogously to Y

using the Cauchy kernel in place of the Bergman kernel. This is a natural extension of

the fact that for p < 1, the dual of Hardy space H p is isomorphic to that of a Bergman

space. Thus it appears that there is enough intrinsic structure to X so that the space begs

for further investigation. Clearly the product of a function in X with the weight function

ln-l(1 - ]z]) -1 belongs to/ :~176 and is in fact continuous on D. We show that in fact

these products axe uniformly continuous with respect to the pseudo-hyperbolic metric on

D and consequently continuously extendable to the union of non-trivial Gleason parts of

the maximal ideal space of H ~176 However, it turns out that these products do not always

extend continuously to all of the maximal ideal space of H ~176

In the second section we use a unified duality approach to give a representation of

functions in X (and X0) as bounded (and compact) multiplication operators (and conse-

quently Toeplitz and Hankel operators) on the Bergman space s Most of the well-studied

representations of multiplication operators concentrate on symbols in the algebra gener-

ated by H ~176 and ~oo. As we show at the end of the first section our approach needs

to be different. We conclude this section by showing that among the compact operators

discussed here, there is no non-zero Hilbert-Schmidt operator.

In the final section we discuss some open problems concerning sub-algebras of/:~176

in the same vein as the interesting questions raised pertaining to the subalgebras of L/

discussed in [4]. It appears that there axe direct links in [10] to our work which go beyond

mere analogy.

Ghatage and Sun 215

N o t a t i o n .

D denotes the open unit disk and dA denotes the normalized Lebesgue measure on D.

f~ = { f , f : D ---* C, f analytic and I n If(z)]2dA(z) < o~} (1 < p < c~) is the Bergman

1 space and for ]AI < 1, kx(z) = (1 - ~ z ) 2 is the Bergman reproducing kernel. The Bloch

space B consists of analytic functions on D which satisfy the growth condition sup (1 - Izl<l

Iz]2)]g'(z)l < 0o and the little Bloch space B0 consists of those functions in B for which

lim (1 - Izl2)]g'(z)] = 0. As a simple consequence of the well-known fact tha t (s = Izl-~l

B[3, Section 2] it follows tha t whenever g �9 B we have, Ig(a)l = I(g,k~,}] < ]]k,~]]lHgl]B.

Consequently, if X = { f , f : D --~ C, f analytic and ] f (a) l ]lka[l~ -1 < c < oo V a �9 D}

then clearly B C X. Similarly if X0 = { f E X, lim If(a)] IIkaH~ -1 = o} then 130 C X0. If I~1--1

OO

Y = { f , f = ~)-'~ankx,, where E lanl like-Ill < c~} with norm JlfH* = inf ~-']~ lanl like, Ill n = l n>_l

where the infimum is taken over all such representations of f , then Y is complete and we

recall that X3 - Y and Y* = X; with (f, g) = f o f (z)y(z)dA(z) whenever f �9 Y and

g �9 X . We write rj(z) = llkzlll 1. For more details and proofs see [7].

H ~176 denotes the Banach algebra of bounded analytic functions on D and .A/I denotes

its maximal ideal space. U may be defined as the C*-sub-a lgebra of / :~176 generated by

H ~176 and ~oo, or the sub-a lgebra generated by bounded harmonic functions or the algebra

C(.Ad), i.e., the algebra of functions of D which can be extended continuously to .s For

m �9 .L4, P(m) is the Gleason par t corresponding to m and G is the union of all non- t r iv ia l

Gleason par ts of .L4\D. For more detailed definitions and prel iminary results see [6].

S e c t i o n 1.

In this section, we give another description of X as p a r t of a Banach space triple

216 Ghatage and Sun

(X0, Z, X) where Z is an appropriately chosen subspace of H 1. Unfortunately the func-

tionsl (., .> which defines the duality involves integration on D with respect to the area

measure. It would certainly be more appropriate to define a functional which involves

integration of boundary values of H 1 functions on the circle.

Next, we look at the functions {r/h, h E X} C/ :~176 We show that every function

in this space is uniformly continuous with respect to the pseudo-hyperbolic metric and

continuously extendable to the subset G of .~4.

We give an example of a function g E X such that r/g ~ U .

Before proving the first proposition we note some well-known facts about H 1 and the

Cauchy kernel. Recall that t t l = {f, f : D ~ C is analytic and sup If(rei~ < e~}. r < l

1 If f~(z) = (1 - ~ z ) ' a E D, is the Cauchy kernel, then III.I1~1 = c ln0 - I~l~) -1 ~

Ic~l ~ 1. There is more than one way to see this but the most straight-forward proof uses (XI

the Taylor expansion of f ( z ) = (1 - z ) - 1 / 2 = ~ anz" with a, ..~ c/x/~. We are thankful n=0

to Sheldon Axler for bringing it to our notice.

Define Z = { f = ~ a , f ~ . , ~-~ [a,[ IJf~.lllt~ < oo} with [[fl[, - i n f ~ l a , I life. ill, I I

the infimum being taken over all such representations of f . Clearly if f E Z then f E H 1

and ]]f[[H ~ < []f][,. As in [7] it can be shown that Z is complete and the dual and predual

of Z can be described as follows.

P r o p o s i t i o n 1. Z* = {h,h, analytic on D, sueh that sup [[f~IIH~]H(a)[ < oo [~1<1

whereH(a) -- h(~)d~} and Z, = {h,h analytic such that IH(~)I III~11~ ~ 0 as

Moreover (f,g} = ~ f(z)-h(z)dA(z) for f E Z and H E X. i

1}. ---+

P r o o f . If h E Z* then it is trivial to check that it induces a bounded linear functional

Ghatage and Sun 217

on Z. As a ma t t e r of interest we note that the Taylor coefficients of h(z) = ~ h , z " satisfy

the condition sup ~ h , a " N < I z..., ~ < er Conversely if L E Z*, then define H(a) = L( f~ ) and

h(z) = (zH(z) ' . The details involving analyticity of H can be repeated almost verbat im

from [7, Prop. 3]. Similarly there is an obvious correspondence between Z. and X0 and

once again the result follows in a manner analogous to [7, Prop. 2].

R e m a r k 1. From [9, Thm. 3.4] we see that for 0 < p < 1, H p has the same dual space

as the Bergman space A 1'1/p-2. The previous proposit ion shows that a similar si tuation

holds not for H 1 and s but ra ther for two special subspaces of H 1 and L:~.

Before proving the next proposition, we recall that if r/(a) = [[ka[[~ -1 then ~(a) =

[a[21n - I ( 1 - [ a [2 ) -1 for a ~ 0 and r/(0) = 1. For a proof of this computa t ion see [7, Lemma

1]. Clearly ~/(a) = r/([a[), 7/is a decreasing function of [a] and depends continuously on

a; q(a) ~ 0 as la[ --* 1. Hence {~h, h E X} _C C(D) N s176

P r o p o s i t i o n 2. If ~ E X, then qT is uniformly continuous on D with respect to the

pseudo-hyperbolic metric.

P r o o f . We denote the pseudo-hyperbol ic distance by p and recall tha t p(z, zl) =

Z - - Z 1

1 - ~zl " For e > 0 let 5(e) = e 2. We only need to prove tha t lira sup sup [~/(z)T(z) - e---*O zED p(Zl,Z)<~

(zl) (zl)l = 0. (,)

For any z0 E D, we define R(zo) = e(1 - [ z0 [ 2) and refer to it as R in the rest of the

argument. Let D(z0, R) = {z, [z - z0 [ < R} be the Euclidean disk. By a s tandard appli-

1 [ _ ~ ( ( )~ ( ( )dA( ( ) l < cation of the mean-va lue theorem, we have I~(z0)~(z0) - ~ Ju(zo,R)

II~ll** fD [ 1 - r](z~ I ~rR2 (zo,R) q~-) dA(~). As is a radial function, without loss of generality we may

218 Ghatage and Sun

1 assume that z0 e R and �89 < zo < 1. Clearly ~?(z) ~ In -1 1 - Izl ~" So if z, Zl �9 D(zo, R),

ln[1 - (z0 + R) 2] s u p r](Zl) ~ - - "R ~ e ( 1 - I z 0 1 ~ ) , so 1.h.s. - - * 1 u n i f o r m l y a s Iz01 -~ 1.

zl,zeo(zo,R) ~?(z) ln[1 - (z0 R) 2]

Similarly it can be shown that inf ~l(Zl)/rl(z ) ~ 1 uniformly as [z01 ~ 1. Thus z,zt ED(zo,R)

1 /D ~/ ( ( )~(~)dA(( ) = II,PIl**" o(1) uni fo rmly as Izl---, 1. , l ( ;o)~(zo) - ~ (;,o,~)

It follows that,

1 /D q(()~o(()dA(() I , ( zo )~o(zo) - ,7(z~)~o(~)l _< I1~o11... o(1) + zc2R2(zo ) (,o,R(~o))

2 ~d term <_ ~ (.o)-D(.~) (~)

-< I1~11.. [ID(zo)\D(zl)l + ID(z~)\D(zo)l]

2e 1 - Tozll __

By hypothesis p(zo,zl) < e 2, hence ]z0 - z , I < d[1 -w0zl ] . It follows that

I - - ~ o z l < ( I 4 - I ; o l ) + I;olp ~1 - 5oZ l l - 1 - Izol "

2 This implies [1 - ~0zl[ < and we have

1 - [ z o l

1 , 7 ( z o ) ~ ( z o ) - ~ ( z 1 ) ~ D ( Z l ) [ ~-~ I 1 ~ 1 1 . , o ( 1 ) + 4~ll~l l**

(1 + [z0[)(1 - e2)"

This implies the result since y(z)~(z) is uniformly continuous on any compact subset of

D.

C o r o l l a r y 1. I f h E X then ~h can be extended continuously to the union of non-

trivial Gleasen parts.

Ghatage and Sun 219

P r o o L The hyperbolic distance between two points is given by r =

log 1 + p(zl, z2) Thus a continuous function on D which is uniformly continuous with 1 - p(zl, z2)"

respect to the pseudo-hyperbol ic metric is clearly uniformly continuous with respect to

the hyperbolic metric. The proof of the Corollary now follows verbat im from the proof

given in [5, L e m m a 15].

We end this section with an example of a function g E X such tha t ~g ~ U. We do

this by showing tha t rlg(z ) does not have non-tangent ia l limits as Izl ~ 1.

The construct ion is based on the following elementary propert ies of r /which we list

first.

1. ~(z) .[ 0 as Izl-~ 1.

2. ( l - x ) 1 / ~ - - + l l e a s x $ O +.

3. For any r E (0, 1), 7--1(1 -- X)r 1Ix -'-+ 0 as X -'-r 0 +,

As a first step, inductively we construct an increasing sequence of positive integers

{nk} satisfying the following properties: (rk = I -- 1 )

'*k 1 / e a s k --~ oo . (b)

(r , - - l ( r l+ l ) r~ '+ ' < 1/6 2.

The details of this construction are s tandard and are left to the reader.

We now define g(z) = E ~-l(rk)znk" It is immediate from (c) t h a t ~---~(~- l (rk)) l /n i k_>0

_< 1 and clear f rom (1) tha t q - l ( r k ) ~ oc. Thus g is analytic on D and g ~ B [1, p. 4].

C l a i m 1. ~(z)g(z) E s176176

220 Ghatage and Sun

We may assume z E ( r t , r t+ l ) for t large.

e--1 oo

< , -1 + , -1 I k=0 j=t

Note that the first term is hounded by et (by (a)), so we concentrate on the second one.

If j > t + 2 then ~(z)~-l(rDIzl"J < ~(re)~-l(~j)~L1. So by (c) ~ ~(z)~-l(~e)lzl"~ j >e-4-2

71-2 ,K 2 --~-rl(re ) < --~. I f j = g then rl-l(rt)rl(z) < 1. Hence we only need to prove boundedness

1 T.I-- rl.~.l

of 7l(z)rl-'(rt+t)N. We do this by showing that the function T(r) -- ln-~---r) defined on

(rt, re+l) satisfies ~ ' ( r ) > 0 on (re, re+l). (This particular computat ion is trivial and so

we omit it.) It follows that sup ~(r) is achieved at one of the end-points. As q is r E ( r t , r t + l )

decreasing and r < 1, we have ' that s u p s ( r ) < c~. This completes the proof of Claim 1.

C l a i m 2. sup lrl(z)g(z) - z 'kl --+ 0 as k ~ cr

Since ~/(z) = ~(Izl), the 1.h.s. _< 7/(rk) ~ r/-1 (rj)Jz "i I. Splitting this series as in the proof j:~ k

k-1

of Claim 1, we see that rl(rk)~f'~rl-l(rj) < ek and rl(rk) ~ rl- l(rj)rj_l < 7r2/6 rl(rk) j = l j=kh-1

by (c). The claim is proved since ~(z) --+ 0 as Izl ~ 1.

Now if h(e i#) is the non-tangential (a.e.) limit of rl(z)g(z ) then by (b), Ih(eie)l - ! - - e

a.e. As 7/E C0(D), this forces the anlytic function g to have a non-tangent ial limit of oo

at almost every point of 0D, which is a contradiction. [See 5, end of proof to Corollary

ll].

R e m a r k 2. We remark that the previous proof made a strong use of the choice of

unbounded coefficients. So a natural question is: Does there exist a set of necessary and

sufficient conditions which identify the set {h E X, ~/h E ~/}? Clearly whenever h E X0,

qh E C0(D) C_//. We come back to this again in Section 3.

Ghatage and Sun 221

Conjecture. If ~ -- C*-alg {~X} then ~ D H.

Section 2.

In this section we give a representation of functions in r/X as multiplication operators

on L1. Our approach differs from the one in [10, Cor. 8] in that we do not start with a

symbol which is assumed to be in L~176 In order to use the generalized duality approach

effectively, we first prove the following theorem.

T h e o r e m 1. Suppose h E s Then the Toeplitz operator Tlhnl 2 is compact if and

only if h E Xo.

Recall that a Toeplitz operator T~ with symbol ~ is defined by T~f = P(fqv) whenever

f E s and f ~ E s and a Hankel operator H~ induced by ~ is defined by H~f =

(1 - P)(f~) whenever f E L~ and f ~ E L2(D). (as always, P is the orthogonal projection

from s to L1-) The multiplication operator M~ is defined by M~f = f~. We have:

H~H~ = TI~I~ - T~T~, whenever both sides of the equation make sense, in particular

whenever ~ E L~176 Moreover M~ = T~ + H~ and T~ and H~ have orthogonal ranges.

Before proving the theorem, we recall some basic facts from [8].

1. A Carleson square Sh is a set of the form Sh ={pe ie : 1 -- h < p < 1, 80 <

8 < 80 + h}. For any positive measure # on D there exists a constant C > 0 such that

/D Ifl2d~ ___ c / v Ifl2dA for all f ~ z:~ if and only if #(Sh) = O(h 2) uniformly o v e r

80 E [0, 27r].

2. If ~ is a positive function in L~176 and d# = ~dA, then the operator T~ is

compact if and only if p(Sh) = o(h 2) independently of 80.

In order to prove the theorem in somewhat greater generality we make the following

222 Ghatage and Sun

definition.

D e f i n i t i o n 1. Let w(r) be a non-negat ive continuous function on [0, 1] with w(1) =

O. w is said to satisfy the growth condition (G~) i f t'or any small h(> 0), there exJsts

6 > 0 and a constant C(5) (both constants axe independent o f h) such that w(1 - h) _<

c(~) inf w(r). rE[1--h--6h,1 --h+~h]

L e m m a 1. ,7(Izl) satis~es the (a~) condition.

P r o o f . For convenience we write r = Izl and observe that, as r/(r) is a decreasing

function of r,

inf{r}(r), r �9 [1 - h - 5h, 1 - h + ~ih]} = 7/(1 - h + 5h)

and hence y(1 - h) _ y(1 - h) infre[1--h--~h,l--h+6h] r/(r) r/(1 -- h + 5h)

(1 - h) ~ ln[1 - (1 - h + ~h) 21 (1 - h + ~h) 2 ln[1 - (1 - h) 2]

in(1 - ~ )h ,~ as h ~ 0 +

In h

ln(1 - 6) + In h

lnh

--ln(1-~-~---~)+1~1 as h---~O +. In h

In conclusion, we note that the computat ion above makes heavy use of elementary special

properties of the logarithm, viz. log uv = log u + log v.

The next proposition gives a slightly more general form of Theorem 1.

P r o p o s i t i o n 4. I f w is a weight- function ( w H ) = w(z ) satisfying the proper ty ( G~),

then for any r C s Mwr is compact i f and only i f lim Iw(]z])r = o. tzl~l

P r o o f . Suppose r E s and let d# = Iw(z)r It is well-known that dA(Sh) ",

h 2 and hence whenever w ( z ) r --~ 0 as Izl ~ I clearly I~(Sh) = o(h2). Now by (2) T1~r

Ghatage and Sun 223

is compact . The basic relation: H$H~ = Tl~12 - T~T~ valid for ~ E s176176 shows that

Twr and Hwr are compact and hence M , ~ is compact . Conversely suppose tha t M,~r

and hence Twr and Hwr are compact . Thus Tiwcp is compact and hence again by fact (2)

~t(Sh(#)) = o(h 2) uniformly over #. So without loss of generality we may look at a Carleson

square whose arc is centered at 1(6 = 0) and whose lateral length is 2h. If D(1 - h,6h)

is the Euclidean disk inside Sh then by the mean value theorem applied to the analytic

function r we have,

1 /D W(1 -- h)Ir Iw(1 - h ) r - h) l 2 <_ z ~ T h 2 (1-h,$h)

Now by the (Ga) contition we obtain

c2(6) [ c2(~) Iw(1 - h)r - h)[ 2 < ~ Js [w([zDr < r-~-~l~(S2h) --" 0

2 h

by hypothesis. This completes the proof.

In order to extend the representation to ~/X we need to use the fact tha t the map

given by h ~ M,h is bounded below on X0.

Recall tha t for h 6 X, [[h[[** = sup {[h(a)[Tl(a)} I~i<1

P r o p o s i t i o n 5. For h E Xo, [[hl[** < C[[M~h][ where M~h : s __~ s and C is

independent of h.

P r o o f . Suppose g . E X0 for n _> 1 and [IM~g. - Mng, [[ --+ 0 as n, m --+ oo. It follows

/ o 17/[2[gn " gm[2dA --~ 0 as n,n ~ oo. As T/is bounded below on compact subsets that

of D, by a s tandard argument we may write, g(z) = lirnoog,,(z ) and conclude that as the

limit is uniform on compacta , g is analytic on D. It only remains to prove tha t g E X0.

224 Ghatage and Sun

This rests on the following claim. Claim: li--~sup [ lq(g. -gm)l 2dA = ~ If not, as in

the proof of Proposition 3 of [8], we conclude that for any N we may find n,m > N such

that [iM.~. - M~m ii > ~ > 0. This eontradiets our hypothesis and the claim is proved.

Now by the argument given in the proof of Theorem l, lim(~Ig. - gml)(z) --+ 0 as Iz[ --+ l

and hence lira 71(z)g(z ) -- 0 proving that g E X0. It is clear that sup [r/g. - ~gl --+ 0; }zl - ' 1 I~ l< l

completing the proof of the proposition.

Next, we extend the representation h ~-~ M,~h to all of X. First we recall that if H~

and/-/2 axe Hilbert spaces then Soo consists of bounded operators B from HI --+//2 such

that B * B is a compact operator on H1; the Schatten class Sp consists of those operators

B in So~ for which eigen-vatues of the positive compact operator B * B belong to e 2p. It is

well-known that S ~ = $1. For more on this and general duality, see [11]. For proof of the

facts: X~ = Y and Y* = X see [7].

L e m m a . Is L : Xo -~ Soo (s --+ s denotes the map h ~-+ Mnh then the dual map

L*: S , ( s 2 ~ s ~ Y is onto and hence 5(Y)1 = {g E Y, Ilgll < 5} c_ ~*((Sl)I ) for s o I i 2 e

5 > 0 .

P roo f . This is a simple consequence of Proposition 5 and the open-mapping theorem.

T h e o r e m 2. For r in s r E X i f and only i f M~r is bounded and []M~[[

defines an equivalent norm on X .

Proof . I f , r E X then r/r E s176 and obviously M,,~ is bounded; and [[Mnr

IlrgblloQ = II~b[l**. Conversely suppose r E /:2 and M,~r is bounded. In order to prove

that r E Y* we look at sup{[{g,r g E Y, [[g[[. _< 1} which by Lemma is equal to

Ghatage and Sun 225

sup{[(L*A,r I[AIl,~ < c} = sup{I(A,Lr [[A[[,~ < c} = sup{t(A,M,~,p)l, IIAIIt~ _< c} <

c[[M~r It follows that r E Y* = X; and {[r _< c[[M~r

P r o p o s i t i o n 6. For r E Xo the operator M~r belongs to the Hilbert-Schmidt class

i f and only i f r - O.

Pro o f . Suppose r E X0 and M~r E 5'2. It follows that Tinr E S1. By [2, Prop. 3.5] .

tr(Tl~r ) = fo(l~?r k),)~:2dA(A)

=/o f. dA(A) I 1 _ ~ 1 4 dA(~).

Hence

fD 1~(0r . . . . . M = 0 - - I ~ - ~ anL~) < oc.

On the other hand, if h < 1/4 then for [z[ < ~ < 1/4 the Cauchy integral formula gives;

r = fl,J---1-h r - z ) d t .

Hence

I r < - - 1 ]o 1 - h - g 1r - h)elO)12dO for Izl < 1/4

< -- :.__ /1-- 4 /27r I r

- h 2 ( 2 - h ) e M as ~? is decreasing < ( 1 - - 5 ) h ~?(1 3)

(1 h 2 2 1 -_~) ( 2 - h ) h In - 4 0 as h--~0

< 1- - -h- -~ 3( 2 h)

r is analytic and hence r ~ 0.

226 Ghatage and Sun

S e c t i o n 3.

The last section contains some remarks and open question.

In section 1 we constructed a function in T/X which does not belong t o / / . In this

as in the example given in [5] the existence of non- tangent ia l l imits plays a crucial role.

This observation becomes more interesting in light of the following example. If r =

r/(z) ln(1 - z) -1 then lim r = 0 unless z --* 1. If r e L/ then a s t ra ight- forward Izl--.x

computa t ion (the details of whixh we omit at this stage) shows tha t r E COP. In this case,

for all ~ E H ~ , ~ r E AOP with ~r = 0. In particular, how does this phenomenon

relate to the question posed by Axler-Gorkin in [4, Section 6]: AOP = H ~ + COP?

As every function in r/X is uniformly continuous with respect to the pseudo-hyperbol ic

metric and hence continuously extendable to the union of non-tr ivial Gleason parts of M ,

it makes sense to make the following definitions and raise the following questions.

Notation: Mn = the maximal ideal space of ~.

Q u e s t i o n 1. If

AOPr = {g E fl; g IF(n) e H (D) for all in M \D}

and

COP(,1) = {g E fl; g I P(,,) = constant for all m E Az/a\D}

is it t rue tha t AOP(~) = H~ + COP(T)? See [4].

Q u e s t i o n 2. Let Ta = {Tg, g E fl} and C is the commuta te r ideal of T~. Obviously

Tn/C is an abelian C*-algebra. How can the algebra be realized as a quotient of C(M~)

by one of its subalgebras? See [8].

Ghatage and Sun 227

R e m a r k 3. Karel Stroethoff has communicated to us that a direct computation using the

Berezin transform of Ihr/I 2 for h E X gives a proof of our Theorem 2 without appealing to

duality. Also Proposition 4 is valid for any weight function w on D (not necessarily radial)

which is bounded on compact subsets of D and satisfies the following growth condition: for

some 0 < s < 1 there exists a positive constant c such that inf{w(z) : z �9 D(A, r)} _> cw(~)

whenever s < ]A[ < 1. (Here D(,k, r) denotes the pseudohyperbolic disk with centre A and

radius r.) Further the Berezin transform argument can be used to show that if h �9 X then

M,sh �9 Sp (p >_ 2) if and only if h - 0. This considerably strengthens our Proposition 6.

We are grateful to him for his kind interest.

R e f e r e n c e s

[1] J. Anderson, Bloch Functions: The Basic Theory, Operators and Function Theory,

edited by S. Power, D. Reidel, Dordrecht(1985), 1-17.

[2] J. Arazy, S. Fisher, and J. Peetre, Hankel operators on weighted Bergman space,

American J. of Math. 110(1988), 989-1054.

[3] S. Axler, Bergman Spaces and Their Operators: Surveys of Some Recent Results in

Operator Theory, Vol. I, Pitman Research Notes in Mathematics 171(1988), 1-50.

[4] S. Axler and P. Gorkin, Algebras on the disk and doubly commuting multiplication

operators, Transactions of AMS, Vol. 309, No. 2 (1988), 711-723.

[5] S. Axler and K. Zhu, Boundary behavior of derivatives of analytic functions, Preprint.

[6] :I. Garnett, Bounded Analytic Functions, Vol. 96 in Pure and Applied Mathematics,

Academic Press, (1981), New York.

228 Ghatage and gun

[7] P. Ghatage and S. Sun, A Lueking-type subspace of/31 and its dual, to appear in

Proceedings AMS.

[8] G. McDonald and C. Sundberg, Toeplitz operators on the disc, Indiana Univ. Math.

J., 28(1979), 595-611.

[9] R. Rochberg, Decomposition theorems for Bergman spaces and their applications,

Operators and Function Theory, (edited by S. C. Power), D. Reidel, Dordrecht (1985),

225-277.

[10] K. Strothoff and D. Zheng, Toeplitz and Hankel operators on Bergman spaces,

Preprint.

[11] S. Sun, Duality and Hankel operators, To appear in the Journal of Integral Equations

and Operator Theory.

Submitted: June 21, 1990

Department of Mathematics Cleveland State University

Cleveland, Ohio 44115

Department of Mathematics Sichuan University

Chengdu, 610064 Peoples' Republic of China