Foiaş, Ciprian

Pearcy, Carl

Szőkefalvi-Nagy, Béla

English

Foiaş Ciprian

Pearcy Carl

Szőkefalvi-Nagy Béla

University of Szeged

Acta Sci. Math., 42 (1980), 201—204-The functional model of a contraction and the space L1 CIPRIAN FOIA§, CARL PEARCY and BÉLA SZ.-NAGY The present Note is a straightforward continuation of the recent paper [I].. Indeed, we have noticed subsequently that, under slightly changed assumptions, the results of that paper can be extended from the factor space U\H£ to the space D itself, and "localized" on parts of the unit circle C. The ingredients of these extensions are mostly taken over, with some changes,, from the paper [I], and so are the notations and the terminology. When referring to a specified lemma or formula of [I] we indicate it by the subscript I. Applica-tions to the invariant subspace problem are to be given later. 1. Let us begin with some lemmas requiring little changes with respect to [I].. Lemma 1. If {a„} converges weakly to 0 in then for any (p£H2 and h£9y we have \\(cpoan)-h*\\Li - 0 and \\h-((poa„)*\\Li - 0 as (This is a strengthening of Lemma 3,, where only convergence in the factor space V-\H\ was established.) Proof. For any h,k£$> the function k-h* is the complex conjugate of h • k* so they have the same norm in L1. Therefore it suffices to prove the first convergence^ Now, by (4.3), and (4.7), we have \\((poan)-h*\\Li S ma^h^^ + li^cpa^hM* - 0 as n Lemma 2. For any and we have |\(ijtoa) • ((pod)* - rj/ip ||a|||, |Li S ||tM|H*(et) ll[0>a] + ||h»<«)+ \\[@*M+ lla^e) Mh»№í)-(This takes over the role of Lemma 4,, with the unpleasant difference that here we have to increase the right hand side of the inequality by a second term.) Proof. It readily follows from (4.2)i and (4.3), that (il/oa)-((poa)* = . ^ M S , - O K 0 [ 0 * < H + ) e , - ( 0 [ 0 * « M + ! cpa)^ Received September 30, 1979. 202 Ciprian Foia$, Carl Pearcy and Bela Sz.-Nagy where [ - ] _= [ • ]—[ • ]+ ; hence, \\(фоаМ<роаГ-Ф<рЫ%.\\*& Since lit • ] — — l i t • ]|li,»((j) and since 0* is also contractive, the proof is done. Lemma 3. Suppose <£„, is (countably) infinite dimensional, and let h, •<Pi> *• • 3 (pT, Фи •••> Ф,£Н2 and e>0 be given. Then there exist h',k'd& such that ( h + w + k y - h . v - z t j v j ш i\\Ф1\\нЛ\ч>М-ЫФ ] )+пв( (р ] ) )+^ . 1 L1 1 PT si 2 ИФЛнг, wrs 2 Ш\2н>-i i Remark. One can choose h', k' even to run over sequences A(n), <n = 1,2, ...) such that, for every /€§, й ( лМ* and kw • I* tend to 0 in L 1 as n-~ Proofs. Almost identical with those of Lemma 5, and Remark,, by using Lemmas 1 and 2 in place of Lemmas 3| and 4,, and applying inequality (5.3), both to (pj and фj. 2. More essential change is needed with Lemma 2,. Its role will be taken by Lemma 4. Given a subset S of the open unit disc D={1: |Я|<1} let s be the set of non-tangential limit points of S on the unit circle C. *) Then for any fdV-(s) and s>0 there exist ..., and cl5 ...,c„£С such that (1) f - 2 c j P t < 6 and Z\cj\ ^ ll/llz.4s), ¡Ms) 1 where P^ is the Poisson kernel function on C corresponding to the point p(£D), i.e. Proof. Suppose there exist f0£L1(s) and e0>0 for which the assertion does not hold, i.e. such that the open ball G in L 1 ^ ) with centre /0 and radius e0 is dis-joint from the set X of all finite linear combinations with Hj£S, cfiC, and 2 lc7'l —ll/ollxi(i)- Since both G and X are convex, and G is open, there exist, by the Hahn—Banach separation theorem, a function g0€Lm{s) (the Banach dual of L 1 ^ ) ) and a real number a such that <3) Re fhg0dm s a < R e ffg0dm x ) For any ScZ>, the corresponding set s c C is a Borel set, indeed an Fa6a. The functional model of a contraction and the space V- 203 for all hdX and /£G (in particular for f=f0); m denotes normalized Lebesgue measure on C. Thus if we set go 0 0 = / g o i e ^ P ^ d m ( j i i D ) s and observe that WPJLHS) ^ \\PJN = 1, and hence, W M M ^ 6 X, the first inequality in (3) shows that (4) ll/oll2.i(S)lfoG*)N « for all p i S . Since Jo is a bounded harmonic function on D, by the Fatou theorem we infer from (4) that ll/olli.i(s)l£b(e")l a almost everywhere on s, so that Re f/ogo dm m ||/0Hi,iw ||gollL~<s) S s This contradicts the second inequality (3). The proof of Lemma 4 is complete. 3. In the sequel the functional t]9(cp) defined in [I] will again play a basic part. Let us recall, in particular, that for (p=p„, where P, (ei') = (l-iiei')-1 (l*<LD), we have f foOg = inf 110(̂ *19111, where 91 runs through the family of subspaces of of finite codimension; cf. (2.6),. For any number 5, consider the subset (5) S> = {(i(iD: of D, and the corresponding set ss of non-tangential limit points of S3 on C. We are going to prove the following substitute for Lemma 5,: Lemma 5. Suppose (S^ is (countably) infinite dimensional and suppose / C L 1 ^ ) and h,k£&, and also e>0 are given. Then there exist h',k'£S$ such that \l(h + h').(k+kT-h'k*-f\\LHsa) s 29||/||Lt(Ss) + 2s, P'll, WW S ||/||tl(S9). Proof . By Lemma4 there exist ..., p„£S and c t , . . . , c„€C satisfying (1) (with s=s3). One can obviously assume that c^0 for all j, so we can set <Pj = |c//2(l - [lijm.j, "Ay = (sgn Cj) . <Pj (y'= 1,2,..., n). Then we have tjVj = CjPN, UjWh = \№i\\H* = \Cj\ 204 Ciprian Foia$, Carl Pearcy and Béla Sz.-Nagy : The functional model and the space Lx so that by Lemma 3 we obtain h', such that (h + AO • (k + kT -h • k* - J Cj S 29Z\cj\ + s L\s») 1 and 1 Taking also account of (1) we conclude the proof. 4. Now we are ready to state the following: Theorem. Suppose &(?•)} is a contractive analytic function, with sep-arable (E+ and with dim = Suppose that for some 9, , the set ss of non-tangential limit points of the set S9 (defined by (5)) on C has positive Lebesgue measure. Then for every /6L1(is) there exist h, such that (6) f = h-k* almost everywhere on s9. Proof . As in the proof of Theorem, we choose a number co such that 23-=co<l and consider an /6Z.x(Js) with li/IIL1^) —1- Setting h-1=ha=k_1= =k0=0 (in § ) we show by induction that there exist h„, k„£§) ( « = 1, 2, ...) such that (?) l !/-/v*n* IL (S s )Say and WK-h^.f, \\K~K_A* ^ CO"-1 (n = 0,1, . . . ) . This being obvious for n=0 we assume /?„, k„ to be already found for n—0, ..., q. Setting fq=f-hq-k* and £,=(co—29) ©"/2, by Lemma 5 we infer that there exist hq+i,kq+1£§> such that IIVi• K+1 - K - K - f q \ ^ 29• II/,IIL>(S3) + 2E4 and \\hq+\ — hqW2, ||*g+1-£8||2 ^ H/,11^) S CO«. Then we have l l/-^ + 1 -^ + 1|| L . ( S s )= ||(/g+ hq • k*)—• ^+111^(5») = 23 • co9+(a>—2S)(oq = co*+1, and the proof of (7) by induction is done. By account of (7), the sequences {h„}, {k„} are strongly convergent (in § ) and their respective limits h, k satisfy (6). Theorem is proved. References [1] B. SZ.-NAGY—C. FOIA§, The functional model of a contraction and the space i 1 / H A c t a Sci. Math., 41 (1979), 403—410. INDIANA UNIVERSITY BLOOMINGTON, INDIANA MICHIGAN UNIVERSITY A N N ARBOR, MICHIGAN UNIV. OF SZEGED SZEGED, H U N G A R Y 

The functional model of a contraction and the space L1

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The functional model of a contraction and the space L1

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