On the Toëplitz Corona problem

Publ. Mat. 47 (2003), 489–496 ON THE TO¨ EPLITZ CORONA PROBLEM E. Amar Abstract The aim of this note is to characterize the vectors g=(g1,...,g k) of bounded holomorphic functions in the unit ball or in the unit polydisk of Cnsuch that the Corona is true for them in terms of the H2Corona for measures on the boundary. Let Dbe abounded domain in Cn, the Corona problem is: given functions g1,...,g Nholomorphic and bounded in Dsuch that: ∀z∈D, N  i=1 |gi(z)|2≥δ2>0, find f1,...,f Nstill holomorphic and bounded in Dsuch that N  i=1 figi=1 in D. This was solved for D=D, the unit disk in Cby L. Carleson [8] and it is still open for n>1 for the basic domains namely the unit ball Bnand the unit polydisk Dn. We shall link this question to a question on To¨eplitz operators via the Hp(µ) Corona. 1. Notations We are interested by the basic domains, the unit ball in Cn,D=Bn, in fact any bounded convex domain with smooth boundary D,orthe unit polydisc D=Dn. If D=Dnwe set bD =Tn, the distinguished boundary; if Dis a bounded convex domain with smooth boundary, bD =∂D the topological boundary. Recall that: H∞(D):=fholomorphic in D/f∞:= sup z∈D |f(z)|<∞. 2000 Mathematics Subject Classification. 32A35, 32A36, 47B35. Key words. Corona problem, To¨eplitz operators, Von Neumann Minimax theorem. 490 E. Amar Let Mbe the set of all probability measures on bD and for µ∈M and 1 ≤p<∞let Hp(µ)bethe closure in Lp(µ)ofthe holomorphic polynomials. If µ∈Mand f∈H∞(D) then, with the assumption that 0 ∈D, for any r<1, fr(z):=f(rz)issuch that fr∈A(D):=H∞(D)∩C(D). There is a subsequence of {fr,r<1}which converges in (L1(µ),L ∞(µ)) topology and uniformly on compact sets of Dto a ˜ f∈H∞(µ)∩H∞(D). Hence for a fixed µ∈Mwe can assume that a f∈H∞(D)isin H∞(µ)∩H∞(D). Now suppose that the Corona problem is solvable, i.e. g1,...,g N∈H∞(D) are such that ∃f1,...,f N∈H∞(D) with 1 = f1g1+···+fNgN;wehave, for any polynomial P: P=Pf1g1+···+PfNgN. Let h∈Hp(µ). Then there is a sequence {Pk}k∈Nof polynomials such that Pk→hin Hp(µ), hence: Pk= N  j=1 Pkfjgj; but then Pkfj→hjin Hp(µ), because the fjcan be seen as in H∞(µ)∩ H∞(D). So if the Corona is true then the Hp(µ) Corona is also true for any µ∈M: CHp(µ):∀h∈Hp(µ),∃k1,...,k N∈Hp(µ) s.t. h= N  j=1 gjkj. The aim of this paper is to show the converse. If f:=(f1,...,f N)weset|f|2(z):= N  j=1 |fj(z)|2and fp:= |f(·)|p, where ·pis the Lp(bD, µ) norm and ·∞is the sup norm in D. On the To¨ eplitz Corona Problem 491 Theorem 1.1. Let Dbeabounded convex domain Dcontaining Oand with a smooth boundary or the unit polydisk Dnof Cn,n≥1.Let: g1,...,g N∈H∞(D)and δ>0. The following are equivalent: (i) There exist functions f1,...,f Nin H∞(D)such that N  i=1 figi=1 and f2 ∞≤1 δ2. (ii) Forall measures µon bD, ∀h∈H2(µ), ∃k1,...,k N∈H2(µ)s.t. h= N  j=1 gjkjand k2 2≤1 δ2h2 2. Let g1,...,g N∈H∞(D)besuch that ∀z∈D, |g|(z)2:= N  j=1 |gj(z)|2≥δ2>0, we already know that: if D=Bn,µthe Lebesgue’s measure on ∂Bnand 2 ≤p<∞, then CHp(µ)istrue [2]; if D=Dn,µthe Lebesgue’s measure on Tnand 1 ≤p<∞, then CHp(µ)istrue [10], [11]; if Dis strictly pseudo-convex, µthe Lebesgue’s measure on ∂D and 2 ≤p<∞, then CHp(µ)istrue [5]; if Dis a bounded pseudo-convex domain with smooth boundary, µthe Lebesgue’s measure on ∂D, then CH2(µ)istrue [4]. In the case n=1,D=Dthe unit disc in C,µthe Lebesgue’s measure on T, then CH2(µ)⇒CH∞(D)[12], by an operator method: the commutant lifting theorem of Nagy-Foias. This means that the Corona theorem in one variable can be proved this way, hence there is some hope to prove a general version of the Corona theorem also by this way. 2. Proof of the theorem We already seen that i) ⇒ii); to prove that ii) ⇒i) we shall use the minimax theorem of Von Neuman. The minimax theorem was already used by Berndtsson [6], [7]inorder to get estimates on solutions of the ∂-equation; here the situation and the method are quite different. 492 E. Amar We shall work with N=2in order to simplify notations. Because D is always convex containing 0, we may assume by dilation that the data g:= (g1,g 2) are continuous up to the boundary, provided that the estimates do not depend on it. Let Ω be an open set in Dsuch that Ω⊂D,0∈Ω and let, for >0, Cbe: C:= {(f=(f1,f 2)∈A(D)2,s.t. 1−f·gΩ≤} where fΩ:= sup z∈Ω |f(z)|; this set is clearly convex in A(D)2. Let Mbe the set of probability measures on bD and for 0 <η≤1 let Mη=ηm +(1−η)M, where m is the Lebesgue measure on bD; this is a convex weakly compact set. Let us define Nas ∀f∈C ,∀µ∈M η,N(f,µ):=f2 µ:= f12 L2(µ)+f22 L2(µ). Then Nis convex on Cfor µfixed in Mηand concave, in fact affine, and continuous on Mfor ffixed in C, hence we can apply the minimax theorem [9]: sup µ∈Mη inf f∈C N(f,µ)= inf f∈C sup µ∈Mη N(f,µ);(∗) by (ii) with h=1wehave ∃k=(k1,k 2)∈(H2(µ))2,g·k=1,kµ≤1 δ; because µ=ηm +(1−η)νwe get km≤1 δ√ηhence k∈(H2(m))2;by the very definition of H2(µ) there is a sequence fn∈(A(D))2such that fn→kin (H2(µ))2hence also in (H2(m))2hence fn→kuniformly on compact sets of D;sofor ≤there is a f∈A(D)2with f−kΩ≤  g∞ and f−kH2(µ)≤. Hence we have 1−f·gΩ=k·g−f·gΩ≤g∞f−kΩ≤ which means that f∈C .Wededuce that the left side of (∗)isbounded by 1 δ2hence for any >0, η>0, γ>0 there is a f,η,γ ∈C with supµ∈MηN(f,η,γ,µ)≤1 δ2+γ. Now let a∈Dand νaa representing measure for asupported by bD, then we have with µ:= ηm +(1−η)νa: |ηf,η,γ (0)+(1−η)f,η,γ (a)|=f,η,γ dµ ≤1 δ+γ and with µ=m, |f,η,γ(0)|=f,η,γ dm ≤1 δ+γ; On the To¨ eplitz Corona Problem 493 hence |f,η,γ(a)|≤(1 + η)(1 δ+γ) 1−η; because this is true for any a∈Dwe get f,η,γ∞≤(1 + η)(1 δ+γ) 1−η. Using Montel property we get that there is a f∈(H∞(D))2bounded by 1 δand such that g·f=1on Ω hence, because Ω is open and f·gis holomorphic in D,f·g=1inD. 3. Operator version We shall give an operator version of the previous result strongly inspired by [3], but first we need some definitions. Let Dbe as before and µ∈M; for any function fin L∞(µ) define the To¨eplitz operator Tµ fon the Hilbert space H2(µ)by ∀g∈H2(µ),T µ fg:= Pµ(fg), where Pµis the orthogonal projection from L2(µ)onH2(µ). We can state: Corollary 3.1. Let Dbe abounded convex domain containing 0and with a smooth boundary or the unit polydisk Dnof Cn,n≥1.Let: g1,...,g N∈H∞(D)and δ>0. The following are equivalent: (i) There exist functions f1,...,f Nin H∞(D)such that N  i=1 figi=1 and f2 ∞≤1 δ2. (ii) Forall measures µon bD, N  j=1 Tµ gj(Tµ gj)∗≥δ21. For D=D2, this was proved in [1]; they used a method specific to the bidisc which explicitly cannot work even for D3. Proof: We shall prove that (ii) is equivalent to: (iii) Forall measures µon bD, ∀h∈H2(µ), ∃k1,...,k N∈H2(µ)s.t. h= N  j=1 gjkjand k2 2≤1 δ2h2 2, and then we apply the theorem to be done. 494 E. Amar (ii) ⇒(iii) (same proof as in [1]): Let µbe a probability measure on bD and set Gi:= Tµ gi;by(ii) we get that the operator Q:= G1G∗ 1+ ···+GNG∗ Nis invertible and  Q−1 ≤1 δ2.Wecan define: Fi:= G∗ iQ−1,i=1,...,N; these are bounded operators on H2(µ) and clearly we get: G1F1+···+GNFN=1.(1) Now take ki=Fih,k:= (k1,...,k N); we have k2 2= G∗ 1Q−1h  2+···+ G∗ NQ−1h  2, but  G∗ 1Q−1h  2=G∗ 1Q−1h, G∗ 1Q−1h=G1G∗ 1Q−1h, Q−1h hence k2 2=h, Q−1h≤1 δ2h2, because (G1G∗ 1+···+GNG∗ N)Q−1=1. Together with equation (1) this means precisely that the H2(µ) Corona is true, i.e. (iii) ∀h∈H2(µ), ∃k1,...,k N∈H2(µ)s.t. h= N  j=1 gjkj,and k2 2≤1 δ2h2. (iii) ⇒(ii): Let µ∈M, then by (iii) we have: ∀h∈H2(µ), ∃k1,...,k N∈H2(µ) s.t. h= N  j=1 gjkjand k2 2≤1 δ2h2 2, then Sh:= {k=(k1,...,k N)∈(H2(µ))N: N  j=1 Gjkj=h}is not empty and it has elements of norm less than 1 δ2h2 2;S0is a subspace of the Hilbert space (H2(µ))Nhence there is a unique element k=(k1,...,k N) in Shwhich is orthogonal to S0and hence of minimal norm. Then we get: k2 2≤1 δ2h2 2and, defining Fjby Fjh:= kj,j=1,...,N,we On the To¨ eplitz Corona Problem 495 have: N  j=1 Fjh2 2≤1 δ2h2 2 (2) ∀h∈H2(µ), N  j=1 GjFjh=h.(3) From equation (3) we get: ∀h∈H2(µ),N  j=1 GjFjh, h=h2 2, hence ∀h∈H2(µ), N  j=1 Fjh, G∗ jh=h2 2, ∀h∈H2(µ),h2 2≤ N  j=1 Fjh G∗ jh  ≤  N  j=1 Fjh2  1/2  N  j=1  G∗ jh  2  1/2 . Using equation (2) we get: ∀h∈H2(µ),h2 2≤1 δh2  N  j=1  G∗ jh  2  1/2 , hence ∀h∈H2(µ), N  j=1  G∗ jh  2≥δ2h2 2and the corollary. References [1] J. Agler and J. E. 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