Journal of Functional Analysis 191, 224–240 (2002) doi:10.1006/jfan.2001.3851 Norm Approximation by Polynomials in Some Weighted Bergman Spaces1 Ali Abkar Department of Mathematics, Imam Khomeini International University, P.O. Box 288, Qazvin 34194, Iran; and Department of Mathematics, Institute for Studies in Theoretical Physics and Mathematics, P.O. Box 19395-1795, Tehran, Iran E-mail: [email protected] Communicated by D. Sarason Received November 2, 2000; revised August 1, 2001; accepted August 1, 2001 The polynomials are shown to be dense in weighted Bergman spaces in the unit View metadata,disk whose citation weights and aresimilar superbiharmonic papers at core.ac.uk and vanish in an average sense at the brought to you by CORE boundary. This leads to an alternative proof of the Aleman–Richter–Sundberg provided by Elsevier - Publisher Connector Beurling-type theorem for zero-based invariant subspaces in the classical Bergman space. Additional consequences are deduced. © 2002 Elsevier Science (USA) 1. INTRODUCTION We denote by D the unit disk, and by T its boundary in the complex plane. A weight function, or simply a weight,inD is any continuous posi- . p tive function w: D Q ]0,+ [. The weighted Bergman space La (D, w), 0<p<+., consists of all analytic functions f on D such that 1/p 1 p 2 ||f|| p = F |f(z)| w(z) dA(z) <+., La(D, w) D where dA(z)=p −1 dx dy is the normalized area measure on D.If . p 1 [ p<+ , it follows that La (D, w) is a Banach space with the above 2 norm, and for 0<p<1, it is a quasi-Banach space. For p=2, La (D, w) is a Hilbert space of analytic functions with the inner product 2 Of, gP 2 =F f(z) g(z) w(z) dA(z), f, g ¥ L (D, w). La(D, w) a D 1 This research was supported in part by a grant from Institute for Theoretical Physics and Mathematics (IPM), Tehran, Iran. 224 0022-1236/02 $35.00 © 2002 Elsevier Science (USA) ⁄ All rights reserved. APPROXIMATION IN BERGMAN SPACES 225 In case that the weight w is identically 1, the corresponding space is called p the Bergman space, and we write La (D) for it, dropping the indication of the weight. A function u defined on D is said to be superbiharmonic provided that D2u \ 0, where D stands for the Laplace operator in the complex plane: 1 “2 “2 D=D = 1 + 2, (z=x+iy). z 4 “x2 “y2 In this paper, we shall consider a class of (non-radial) weights w satisfying the following conditions: (i) D2w \ 0, the weight function w is superbiharmonic, and > (ii) limr Q 1 T w(rz) ds(z)=0, where ds is the normalized arc-length measure on T. The main result of this paper (Theorem 2.6) states that the polynomials p . are dense in the weighted Bergman space La (D, w),0<p<+ , for weights satisfying the conditions (i) and (ii). The result was motivated by a p question about invariant subspaces in the classical Bergman space La (D). Indeed, the question of weighted polynomial approximation for weights satisfying the condition (i) together with a stronger condition than (ii) was raised by Hedenmalm in [4, p. 114]. The main result of this paper answers Hedenmalm’s question in the affirmative. p Background. A closed subspace M of the Bergman space La (D) is said to be invariant provided that zM … M. Assume that M is an invariant 2 subspace of La (D) with index one; this means that the quotient space M/zM has dimension one. In [6], Hedenmalm considered the following extremal problem, (j) sup{Re f (0) : f ¥ M, ||f|| 2 [ 1}, La(D) where j is the multiplicity of the common zero at the origin of all the func- tions in M. The unique solution to this problem is called the Hedenmalm extremal function for M and is denoted by jM . The extremal functions in the Bergman spaces play the same role as the inner functions play in the Hardy spaces. Hedenmalm further showed that there is a function F 5 . jM ¥ C(D) C (D) such that F F 2 1. jM =0 on T and D jM =(|jM | −1) inside D, F 2 2. 0 [ jM (z) [ 1 − |z| , for z ¥ D. A particularly simple class of invariant subspaces is the one characterized by a zero sequence; a subset L … D is said to be a Bergman space zero 226 ALI ABKAR sequence provided that there is a function in the Bergman space which 2 vanishes precisely on L. Let M be the invariant subspace of La (D) consist- ing of all functions which vanish on L, and let jM be the corresponding extremal function for M. Now, the following question is natural: Does jM generate the invariant subspace M? Hedenmalm [6, Corollary 2.4] obtained the following reduction of the problem. Denoting by [jM ] the invariant subspace generated by jM,he found that 2 [jM ]=jM · A0 (jM), 2 where A0 (jM ) is the closure of the polynomials in the Hilbert space 2 3 2 F F 2 .4 A (jM )= f ¥ La (D): jM (z) D |f(z)| dA(z) < + , D with the following norm 2 2 2 2 2 F F D ||f|| A (jM) =||f|| L (D) + jM (z) |f(z)| dA(z). a D 2 By [6, Theorem 4.2], we know that M … jM · A (jM ). Now, the question if jM generates M becomes the question of whether the polynomials are 2 2 dense in the space A (jM ). We will show that the norm in A (jM ) is comparable to 2 2 F 2 ||f|| g =|f(0)| + wjM (z) D |f(z)| dA(z), D 2 2 F where wjM (z)=(1− |z| ) + jM (z). Hence the polynomials are dense in 2 A (jM ) if and only if they are dense in the weighted Bergman space 2 La (D, wjM ). It follows that jM generates M if the polynomials are dense in 2 the weighted Bergman space La (D, wjM ), where the weight function wjM has the following properties: 2 1. D wjM >0, and 2 2. 0<wjM (z) [ 2(1 − |z| ). As an application of Theorem 2.6, we prove that jM generates the invariant subspace M (see Theorem 3.1), a result which was proved in a similar fashion by Aleman, Richter, and Sundberg (see [2, Propositions 2 4.4, 4.5]). Indeed, they proved that the polynomials are dense in A (jM ), 2 2 2 so that A0 (jM )=A (jM ) and hence [jM ]=jM · A (jM ). This answers the question of polynomial approximation for the weights wjM considered above. They showed that F F 2 F F 2 jM (z) D |f(rz)| dA(z) Q jM (z) D |f(z)| dA(z), as r Q 1, D D APPROXIMATION IN BERGMAN SPACES 227 the proof of which uses the representation formula F F 2 jM (z)= C(z, z) D |jM(z)| dA(z), z ¥ D, D whereby C(z, z) is the biharmonic Green function for the operator D2 in the unit disk: z−z 2 C(z, z)=|z − z|2 log : : +(1 − |z|2)(1−|z|2), (z, z) ¥ D × D. 1−zz¯ Weights w satisfying the conditions (i) and (ii) need not have as simple a F representation as the function jM above. Nevertheless, according to a result due to the author and Hedenmalm [1], the weight function w can be represented by (1.1) w(z)=F C(z, z) D2w(z) dA(z)+F H(z, z)dm(z), z ¥ D, D T where dm is a positive Borel measure on T and (1−|z|2)2 H(z, z)= , (z, z) ¥ T × D, |1 − z¯z|2 is called the harmonic compensator (indeed, here we are using a special case of a representation formula established in [1]). To prove the main approximation theorem (Theorem 2.6), we consider fr —the dilation of f by r—defined by fr (z)=f(rz) for 0 [ r<1 and z ¥ D. We then prove that fr Q f in norm, as r Q 1, that is, F p lim |fr (z) − f(z)| w(z) dA(z)=0, r Q 1 D from which the result follows. 2. AN APPROXIMATION THEOREM In [2], the authors studied properties of a function which is closely related to the biharmonic Green function C(z, z), namely, (1 − |z|2)2 (1−|z|2)2 C2(z, z)= , (z, z) ¥ D × D. |1 − z¯z|2 228 ALI ABKAR Among other things, they proved that C2(z, z) has some kind of monotoni- city property. More precisely, for fixed z ¥ D, the function r W rC2(z, z/r) is increasing on the interval (|z|, 1) (see [2, p. 285]). In the following lemma, we shall see that for fixed z ¥ T, the harmonic compensator (1−|z|2)2 H(z, z)= , (z, z) ¥ T × D, |1 − z¯z|2 enjoys the same monotonicity property; the function rH(z, z/r) is an increasing function of r for |z|<r<1. Here, it is important to note that the first argument in C2(z, z) belongs to the unit disk, while the first argument in H(z, z) belongs to the unit circle. As a result, C2(z, z) vanishes for z ¥ T, while H(z, z) does not, meaning that these functions are quite different, that is, if we look at them as functions of z, then H(z, z) is not a constant multiple of C2(z, z).