Toeplitz operators and division by inner functions

First Advanced Course in Operator Theory and Complex Analysis, University of Seville, June 2004 TOEPLITZ OPERATORS AND DIVISION BY INNER FUNCTIONS DANIEL GIRELA, CRIST´ OBAL GONZ´ ALEZ, AND JOS´ E´ ANGEL PEL´ AEZ Abstract. A subspace Xof the Hardy space H1is said to have the K-property if for any ψ∈H∞, the Toeplitz operator Tψmaps Xinto itself. This in turn implies that Xalso has the f-property. This means that h/I ∈Xwhenever h∈Xand Iis an inner function with h/I ∈H1. In this survey paper we present a list of subspaces of H1that have or have not the f- or K-property, showing some of the different techniques and methods used in the subject. 1. Introduction Denote by Dthe unit disk {z∈C:|z|<1}, and its boundary by ∂Dor T, indistinctly. Denote also by Hp(0 < p ≤ ∞) the classical Hardy spaces consisting of those analytic functions fdefined on Dsuch that kfkHp:= sup 0<r<1 Mp(r, f)<∞, where Mp(r, f), 0 < r < 1, are defined as, Mp(r, f) = µ1 2πZ2π 0 |f(reiθ)|pdθ¶1/p ,(0 < p < ∞), and M∞(r, f) = max θ|f(reiθ)|. Date: March 31, 2005. 2000 Mathematics Subject Classification. 30D50, 30D55, 47B35. This research has been supported in part by a grant from “El Ministerio de Educaci´on y Ciencia, Spain” (MTN2004-00078) and by a grant from “La Junta de Andaluc´ıa” (FQM- 210). 85 86 D. GIRELA, C. GONZ´ ALEZ, AND J. A. PEL´ AEZ A thorough study of these spaces is done in Duren’s book [10], from where we review some well known properties in order to motivate the definitions that follow. When 1 ≤p≤ ∞,Hpis a Banach space with norm given by k·kHp, and for 0 <p<1 it is an F-space (complete metrizable topological vector space) with distance function given by d(f, g) = kf−gkp Hp. Every function f∈Hphas radial limits almost everywhere on the boundary, and the boundary function of f, defined as, f(eiθ) = lim r→1− f(reiθ), θ ∈R, is in Lp(∂D) and also log |f(eiθ)| ∈ L1(∂D). (From now on, the three symbols Lp,Lp(∂D), and Lp(T), will denote the same space). Besides, the “Hp-norm” (it is only a true norm when p≥1) of fcoincides with the Lp-norm of its boundary function: kfkHp=µ1 2πZ2π 0 |f(eiθ)|pdθ¶1/p . If f∈Hp,f6≡ 0, then the sequence of its zeros {an}, repeated according to multiplicities, satisfies the so called Blaschke Condition: X n (1 − |an|)<∞. This condition for a sequence {an} ⊂ Dturns out to be also sufficient in order to construct an Hpfunction whose sequence of zeros (repeated according to multiplicities) is exactly {an}. For that, we just need to consider, for each n, the self-mappings of the unit disk, bn(z) = z, if an= 0, bn(z) = an |an| an−z 1−anz,if an6= 0, and realize that the infinite product Qnbn(z) converges absolutely and uniformly on each compact subset of D, defining a function B∈H∞, called the Blaschke product associated with {an}, whose sequence of zeros is exactly {an}, and with the further properties that |B(z)|<1 in D, and |B(eiθ)|= 1 a.e.. All this gives rise to a result of F. Riesz that states that any function f∈Hp with f6≡ 0, can be factored in the form f(z) = B(z)g(z), where B(z) is the Blaschke product associated with the sequence of zeros of f, and gis a zero-free function in Hpwith kgkHp=kfkHp. Actually, the previous can be carried further bringing us to a canonical factorization due to Smirnov. Quoting from Duren’s book, observe that the function F(z) = exp µ1 2πZ2π 0 eit +z eit −zlog |f(eit)|dt¶ is analytic in Dand satisfies |F(eiθ)|=|f(eiθ)|a.e. because log |f(eiθ)| ∈ L1. Also, since |f(eiθ)| ∈ Lp, an application of the arithmetic-geometric mean TOEPLITZ OPERATORS AND DIVISION BY INNER FUNCTIONS 87 inequality shows that F∈Hp. Moreover, |f(eiθ)|=|g(eiθ)|a.e. and |g(z)| ≤ |F(z)|in D, where gis the zero-free factor associated with fin the Riesz factorization. Hence, the function S0(z) = g(z)/F(z) is analytic in Dand has the properties 0<|S0(z)| ≤ 1,and |S0(eiθ)|= 1 a.e., This shows that −log |S0(z)|is a positive harmonic function in Dwhich vanishes almost everywhere on the boundary. Thus by the Herglotz representation theorem and Fatou’s theorem, it can be represented as the Poisson integral of a positive singular measure µwhich, by analytic completion, gives S0(z) = eiγS(z), where γis a real constant. The function S(z) is called the singular inner function associated to µ, S(z) = exp µ−Z2π 0 eit +z eit −zdµ(t)¶. Everything together yields the canonical factorization f(z) = eiγ B(z)S(z)F(z). The function Fabove is an outer function for the class Hp, i.e., it is a function of the form F(z) = exp µ1 2πZ2π 0 eit +z eit −zlog ψ(t)dt¶, where ψ(t)≥0, log ψ(t)∈L1, and ψ(t)∈Lp. On the other hand, an inner function is a function I(z) analytic in Dwith |I(z)| ≤ 1 and |I(eiθ)|= 1 a.e.. The above tells us that any inner function Iis uniquely factored in the form I(z) = eiγ B(z)S(z), where γis a real number, Bis a Blaschke product and Sis a singular inner function. The canonical factorization brings us to the following important observation. Theorem 1. If f∈Hp,1≤p≤ ∞, and Iis an inner function such that f/I ∈H1, then f/I ∈Hpand kf/IkHp=kfkHp. The analysis of this property in different subspaces of H1is the central topic of this survey. Definition 1. A subspace Xof H1is said to have the f-property (also called the property of division by inner functions) if h/I ∈Xwhenever h∈Xand I is an inner function with h/I ∈H1. Rephrasing it, X⊆H1has the f-property if, whenever F∈H1,Iis inner and FI ∈X, one also has that F∈X. This notion, of studying whether the outer part of functions in a given subspace of H1remains in the same space, seems to have appeared in the early seventies. We mention especially the works of Havin [27], Korenblum and Koroleviˇc [36] and Korenblum [33, 34] on the subject where a number of subspaces of H1were shown to enjoy the f-property. 88 D. GIRELA, C. GONZ´ ALEZ, AND J. A. PEL´ AEZ Let us say here that Havin proved in [27] that the following spaces do satisfy the f-property: •Λα,n (n= 0,1, . . . , 0 < α < 1), the space of functions f∈H1such that f(n)∈Lipα(D). •Λ0,n (n= 0,1. . . ), the space of functions f∈H1such that f(n)is continuous in Dand f(n)(eiθ) is smooth in the sense of Zygmund. •Ap,1={f∈H1:f0∈Lp(D)}, 1 < p < ∞. •Hp,1={f∈H1:f0∈Hp}, 1 < p < ∞. In view of this, one is driven to think that spaces not having the f-property must be rare. However, the first such example appeared published in 1972, and was given by Gurarii [25] who proved that the space W+of analytic functions in Dwith absolutely convergent power series does not possess the f-property. Before getting deeper into the study of our next question, let us introduce the other type of property we are interested in, and that implies the f-property. That will be called “the K-property”. Notice that any function f∈H1can be recovered from its boundary function by the Cauchy integral formula, f(z) = 1 2πi Z∂D f(ζ) ζ−zdζ =1 2πZ2π 0 f(eit) 1−e−itzdt, z ∈D. This is nothing else but the Szeg¨o projection of the boundary function f(eit). In general, the Analytic Szeg¨o Projection of an L1(∂D) function ψ(eit)∼ P∞ −∞ ˆ ψ(n)eint is defined as, Pψ(z) = 1 2πZ2π 0 ψ(eit) 1−e−itzdt =1 2πZ2π 0 ∞ X n=0 ψ(eit)e−int zndt = ∞ X n=0 ˆ ψ(n)zn, which, in fact, is analytic in D. Notice that, according to the M. Riesz theorem on conjugate functions, the analytic Szeg¨o projection operator maps Lp(∂D) boundedly onto Hp, if 1 < p < ∞. However, this is not true either for p= 1 or for p=∞. Returning to our main stream, observe that if his analytic in D,Iis inner, and h/I ∈H1, then, using that |I(eiθ)|= 1 a.e., (1) h I(z) = 1 2πZ2π 0 h I(eit) 1−e−itzdt =1 2πZ2π 0 h(eit)I(eit) 1−e−itzdt =P(hI)(z). TOEPLITZ OPERATORS AND DIVISION BY INNER FUNCTIONS 89 Definition 2. Given v∈L∞(∂D), the Toeplitz operator Tvassociated with the symbol vis defined as Tvf(z) = P(vf)(z) = 1 2πi Z∂D v(ζ)f(ζ) ζ−zdζ, f ∈H1, z ∈D. Many basic facts about Toeplitz operators can be found in Chapter 7 of [9] and in Appendix 4 of [38], for example. Notice that (1), can be written as h I=T¯ I(h) . Here, of course, the symbol Iis understood as the boundary function associated with I. Definition 3. A subspace Xof H1is said to have the K-property if Tψ(X)⊆ Xfor any ψ∈H∞. From the above discussion we obtain that the K-property implies the fproperty. Also, the fact that the Szeg¨o projection maps Lp(∂D) boundedly onto Hpwhenever 1 < p < ∞implies easily the following. Theorem 2. If 1< p < ∞, then Hphas the K-property. ¤ As we mentioned above, the purpose of this survey is to build a (nonexhaustive) list of subspaces of H1that have or have not the f- or K-property. Actually, we are mainly interested in showing some of the different techniques and methods used in the subject. 2. Representation formulae The Dirichlet space Dconsists of those functions f(z) = P∞ 0anznanalytic in Dwith Dirichlet integral D(f) finite, i.e., D(f) := ³1 πZZD |f0(z)|2dA(z)´1/2≡³∞ X n=1 n|an|2´1/2<∞. Theorem 3. The Dirichlet space has the f-property. Proof. Carleson [8] obtained a simple explicit formula for the Dirichlet integral of a given function finvolving the zeros of fand its boundary values. Namely, he proved that if F∈H2and I=BS, where Bis the Blaschke product with sequence of zeros {an}and Sis the singular inner function generated by the measure µthen D(FI)2=D(F)2 +1 2πZ2π 0 |F(eit)|2ÃX n 1− |an|2 |eit −an|2+ 2 Z2π 0 dµ(ξ) |eit −ξ|2!dt. By means of this representation formula, one obtains easily that, whenever F∈H2and Iis inner, D(F)≤ D(FI), from where we deduce that if FI ∈ D, then so is F, concluding the result. ¤ 90 D. GIRELA, C. GONZ´ ALEZ, AND J. A. PEL´ AEZ Given a nondecreasing weight sequence w={wn}∞ n=1,wn≥0, we define the Dirichlet-type space Dwas the set of functions f(z) = P∞ n=1 anznfor which D2 w(f) := ∞ X n=1 wn|an|2<∞. Dyakonov [11] proved an analogue of Carleson formula for these spaces which gave rise to “orthogonality relations” of the form D2 w(FI) = D2 w(F) + Rw(F, I), F ∈H2, I inner, with Rw(F, I)>0. This immediately yields that the spaces Dwhave the f-property. Let us remark here that Rabindranathan [40] and Korenblum and Fa˘ıvyˇsevski˘ı[35] independently had proved that these spaces in fact have the stronger K-property. Another instance in which a representation formula plays an important role in deciding whether the given subspace of H1satisfies the f-property is in the case of BMOA, the space of H1functions whose boundary values have bounded mean oscillation; and in the case of V MOA, consisting of those BMOA functions with vanishing mean oscillation, also characterized as the closure of the polynomials in BMOA. A good account on BMOA may be found in the survey paper [20]. Theorem 4. BMOA and V MOA have the f-property. Proof. Among the many characterizations of BMOA and V MOA we rest upon the one using the Garsia norm: f∈BMOA ⇐⇒ sup a∈Dµ1 2πZ2π 0 |f(eiθ)|21− |a|2 |1−aeiθ|2dθ − |f(a)|2¶<∞, f∈V MOA ⇐⇒ lim |a|→1µ1 2πZ2π 0 |f(eiθ)|21− |a|2 |1−aeiθ|2dθ − |f(a)|2¶= 0. Thus, if FI ∈BMOA (V MOA) with F∈H1and Iinner, then 1 2πZ2π 0 |FI(eiθ)|21− |a|2 |1−aeiθ|2dθ − |FI(a)|2 =1 2πZ2π 0 |F(eiθ)|21− |a|2 |1−aeiθ|2dθ − |F(a)|2+ ≥0 z }| { |F(a)|2¡1− |I(a)|2¢ ≥1 2πZ2π 0 |F(eiθ)|21− |a|2 |1−aeiθ|2dθ − |F(a)|2. From here it is obvious that Fitself is in BMOA (V MOA). ¤ TOEPLITZ OPERATORS AND DIVISION BY INNER FUNCTIONS 91 Dyakonov [13] has introduced recently the notion of a Garsia-type norm (GTN) on a Banach space of analytic functions in D: Let Xbe such an space and assume that X⊂Hpfor some p∈(0,∞). Write |Hp|for the set of nonnegative functions g∈Lp(∂D) satisfying either log g∈L1(∂D) or g= 0 a.e.; these are precisely the boundary values of the moduli of Hp-functions. Suppose that there exists a mapping Ψ : |Hp| × D→[0,∞] and a function k:D→(0,∞) with the following properties: (a) Ψ(cg, z) = cpΨ(g, z), whenever g∈ |Hp|and z∈D. (b) Ψ(|f|, z)≥ |f(z)|p, for all f∈Hpand z∈D. (c) The quantity (2) Np,Ψ,k(f) := sup z∈D {Ψ(|f|, z)− |f(z)|p}1/p k(z), f ∈Hp, is comparable to kfkXwith constants not dependidng on f(it is understood that kfkX=∞if f∈Hp\X). Then we say that N(·) is a GTN on X. Of course, the first example of a Garsia-type norm is the classical Garsia norm in BMOA and, just as it happens in this space, if Xhas a Garsia-type norm then it satisfies the f-property. Dyakonov proved in [12] that, for 0 < α < 1, the Lipschitz spaces Λα= Λα,0 have a GTN and, hence, the f-property. As mentioned above this was first proved by Havin [27]. Dealing with the perhaps more natural Lipschitz space Λ1= Λ1,0is harder. N. A. ˇ Sirokov [48] proved that Λ1has in fact the f-property. His proof was very involved. Dyakonov [13] has proved that Λ1has a GTN obtaining in this way an alternative proof of ˇ Sirokov’s result simpler than the original one. 3. Duality The idea of Hedenmalm of using a duality argument to study the K-property in different subspaces of H1has been very fruitful. Most of the results in this section are from his paper [28]. Recall that BMOA is the dual space of H1under the pairing hf, gi= lim r→1− 1 2πZ2π 0 f(reiθ)g(eiθ)dθ, f ∈H1, g ∈BMOA. Theorem 5 (Hedenmalm [28]).BMOA has the K-property. Proof. For a given ψ∈H∞, we claim that the Toeplitz operator Tψdefined in BMOA is in fact the adjoint operator of the operator Mψ, multiplication by ψ, which of course, is continuous from H1to H1. Under the validity of this claim, Tψis also continuous from (H1)∗=BMOA to (H1)∗=BMOA, concluding the result. To prove the claim, take f∈H2, which is dense in H1, and g∈BMOA (ψg ∈H2). Then, recalling that Pdenotes the analytic Szeg¨o projection, we 92 D. GIRELA, C. GONZ´ ALEZ, AND J. A. PEL´ AEZ have hf, (Mψ)∗(g)i=hMψ(f), gi = lim r→1− 1 2πZ2π 0 (ψf)(reiθ)g(eiθ)dθ =1 2πZ2π 0 (ψf)(eiθ)g(eiθ)dθ [(ψf), g ∈H2] =1 2πZ2π 0 f(eiθ)(ψg)(eiθ)dθ =1 2πZ2π 0 f(eiθ)P(ψg)(eiθ)dθ    the non-analytic part of ψg is annihilated by f∈H2   , from where we deduce that (Mψ)∗=Tψ, and the claim is proved. ¤ A similar argument can be applied to H1obtaining the following result. Theorem 6. H1does not have the K-property. Proof. Assume by contradiction that Tψ(H1)⊆H1for all ψ∈H∞. Then by the Closed Graph Theorem all these Tψare continuous from H1to H1. So their adjoints (Tψ)∗are continuous from BMOA to BMOA. Let us see the aspect of these adjoint operators. For ψ∈H∞,f∈H2(dense in H1), and g∈BMOA ⊂H2, hf, (Tψ)∗gi=hTψf, gi = lim r→1− 1 2πZ2π 0 P(ψf)(reit)g(eit)dt =1 2πZ2π 0 P(ψf)(eit)g(eit)dt [P(ψf), g ∈H2] =1 2πZ2π 0 f(eit)(ψg)(eit)dt    the non-analytic part of ψf is annihilated by g∈H2   , so (Tψ)∗is the operator Mψ, multiplication by ψ. The continuity of such operator from BMOA to BMOA implies that H∞is a subspace of the space of multipliers of BMOA, but we know that this is false: there exist Blaschke products that are not multipliers of BMOA, not even of the Bloch space (to be defined later). See for instance [37, 19]. ¤ Denote by Athe disk algebra. This is the space of analytic functions in D that admit a continuous extension to D. We can see that Ahas the f-property arguing as in the proof of Theorem 6.3 in p. 78 of [18]: TOEPLITZ OPERATORS AND DIVISION BY INNER FUNCTIONS 93 Suppose f6≡ 0, f=FI ∈ A with Iinner and F∈H1(hence, F∈H∞). Let K={ξ∈T:f(ξ) = 0}. Then Kis a closed set of (one-dimensional) measure zero. Now, if ξ∈T\Kthen |f(ξ)|>0 and, hence, I(zn) cannot tend to zero for any sequence of point {zn} ⊂ Dtending to ξ. Then, using Theorem 6.1 and Theorem 6.2 in Chapter II of [18], we deduce that Iis analytic across T\K and then it follows that Fis continuous at each point of T\K. Notice that we certainly have |F|=|f|on T\K. Set now F(ξ) = 0 for all ξ∈K(since K has measure zero, defining Fin this way at the points of Kcauses no problem at all). Then |F|=|f|on Tand |F|= 0 on K. Since fis continuous on T, it follows that Fis continuous at each point of K. Thus we have that Fis continuous on T. The Poisson representation now implies that F∈ A. On the other hand, using duality arguments, we can prove that Adoes not possess the K-property. Hruscev and Vinogradov [29] proved that the dual space of Ais identified with K, the space of Cauchy integrals of complex Borel measures on ∂D. In the same paper, they also characterized the inner functions that are multipliers of K, which for our purposes it suffices to know that they are not all of them. Theorem 7 (Hedenmalm [28]).Adoes not have the K-property. Proof. The same duality argument as before would show that if Ahas the K- property, then H∞would become subspace of the space of multipliers of K, and, as we have just said, this is not true. ¤ 4. Polynomial approximation Our next aim is studying the K-property in V MOA. We recall that V MOA is the closure of the polynomials in BMOA (see e.g. Theorem 5.5 of [20]). Thus, it becomes interesting to see the behavior of Toeplitz operators on polynomials. Lemma 8. If ψ∈H∞and pis a polynomial, then Tψpis again a polynomial. Proof. Write ψ(z) = P∞ n=0 anznand p(z) = PN n=0 bnzn, then, for almost every θ, ψ(eiθ)p(eiθ) =³∞ X n=0 ane−inθ´³N X n=0 bneinθ´ = N X n=−∞ ³N X k=max{n,0} a|n−k|bk´ | {z } =cn einθ, so that Tψp(z) = P(ψp)(z) = PN n=0 cnznis indeed a polynomial. ¤ With this, we are able to prove the following. Theorem 9 (Hedenmalm [28]).V MOA has the K-property. 100 D. GIRELA, C. GONZ´ ALEZ, AND J. A. PEL´ AEZ Theorem 16. B ∩ Hp(1≤p≤ ∞) does not have the K-property. Actually, adapting the proof of Theorem 10, more can be said. Theorem 17. If Xis a subspace of B ∩ H1containing B0∩H∞, then it fails to have the K-property. Proof. Assume by contradiction that Xhas the K-property. Then for any ψ∈H∞,Tψ(X)⊆X, and hence, by the Closed Graph Theorem they are continuous, and by the hypothesis, Tψ(B0∩H∞)⊆Tψ(X)⊆X⊆ B ∩ H1. Next using the same notation as in the proof of Theorem 15, we have that TIB=B/I =B1/∈ B0. However, since B∈ B0, there exists a sequence of polynomials {qn}that approximate Bin the Bloch norm. So, bearing in mind that TItransforms polynomials into polynomials and that TIis continuous from B0∩H∞to B ∩ H1, we obtain that TIqnis a sequence of polynomials approaching to TIB=B1in the B ∩ H1-norm. This implies that B1∈ B0, which is false. ¤ In the same paper [21] a result is proved which answers the above question. Theorem 18. If 1≤p < ∞then B ∩ Hpdoes not have the f-property. Sketch of the proof. It goes as follows. Let Bbe an infinite Blaschke product in B0, whose sequence of zeros have a subsequence accumulating at 1. Then µ(r) = sup r≤|z|<1¡1− |z|2¢¯¯B0(z)¯¯&0,as r%1. Let B1be an interpolating Blaschke product, subproduct of B, whose sequence of zeros accumalate at 1. Construct now a conformal mapping Ffrom Donto a circularly symmetric and starlike domain with respect to 0 satisfying the following conditions (see [7, Section 3] for the construction of such mappings): F∈BMOA;F(z)−−−→ z→1∞;M∞(r, F) = O³1 µ(r)´,as r→1. From this, obtain that f=B·F∈ B ∩ Hp, (p < ∞), I=B/B1is inner, f/I =B1·F∈H1, but f/I /∈ B.¤ 7. Final comments First of all, we have to remark that, in addition to those that we have considered, many other results have been obtained in this subject. Let us simply mention the works of F.A. ˇ Samojan [42, 43, 44], R.F. Shamoyan [46], N.A. ˇ Sirokov [49, 50, 51, 52, 53, 54] and S.A. Vinogradov [55], as well as, those of Axler and Gorkin [5], K. Izuchi and Y. Izuchi [30, 31] and Gorkin and Mortini [22, 23, 24] where these and other related questions are studied in the setting of Douglas algebras, for further reading. TOEPLITZ OPERATORS AND DIVISION BY INNER FUNCTIONS 101 We would like to thank K.M. Dyakonov and R. Mortini for some conversations on the subject which helped us to improve the paper. We wish also to thank the referee for his/her comments and remarks. Finally, we wish to thank also Alfonso and his students Manuel and Alejandro for organizing such an interseting meeting and for their great hospitality. References [1] J.M. Anderson, On division by inner factors, Comment. Math. Helv. 54 (1979), no. 2, 309-317. [2] J.M. Anderson, J. Clunie and Ch. Pommerenke, On Bloch functions and normal functions, J. Reine. Angew. Math. 270 (1974), 12–37. [3] R. Aulaskari and P. 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