Composition operators with linear fractional smbols and their adjoints

First Advanced Course in Operator Theory and Complex Analysis, University of Seville, June 2004 COMPOSITION OPERATORS WITH LINEAR FRACTIONAL SYMBOLS AND THEIR ADJOINTS MAR´ IA J. MART´ IN Abstract. We characterize all linear fractional maps of the disk into itself in terms of their coefficients. We also prove the formula for the adjoint of a composition operator with a linear fractional symbol acting on the quotient Dirichlet space due to Gallardo and Montes by a method different from theirs. Introduction As usual, we will denote by Dthe unit disk in the complex plane: D={z∈ C:|z|<1}. We will call ϕaself-map of Dif it is a holomorphic (analytic) map in Dsuch that ϕ(D)⊂D. The composition operator with symbol ϕis defined as Cϕf(z) = f(ϕ(z)), for any self-map ϕ. The subject of composition operators has been an active area of research for more than thirty years (cf. [8] and [2]). We will study such operators acting in the quotient Dirichlet space. Denoting by dA the normalized area Lebesgue measure π−1dxdy, the Dirichlet space D is defined as the Hilbert space of analytic functions in Dwhose derivative is of square integrable modulus with respect to dA. For f, g ∈ D with Taylor series around z= 0 given by P∞ n=0 anznand P∞ n=0 bnznrespectively, we define their inner product in Das (1) hf, giD=a0b0+ ∞ X n=1 n anbn. 2000 Mathematics Subject Classification. 47B33, 31C25. Revised October 10, 2004. This work is partially supported by MCyT grant BFM2003-07294-C02-01, Spain. 105 106 M. J. MART´ IN If we want to work in the quotient space D0=D/C, the Dirichlet space modulo constant functions, the inner product is defined by (2) hf, giD0= ∞ X n=1 n anbn. The problem of the computation of the adjoint of a composition operator with linear fractional symbol was first solved by Cowen [1] in the Hardy space. Later on, Hurst [6] using an analogous argument obtained the solution in the weighted Bergman spaces A2 α. In 2003, Gallardo and Montes [3] computed the adjoint of a composition operator acting in the Dirichlet space by a different method from those used by Cowen and Hurst. In this paper we first give a characterization in terms of the coefficients of all linear fractional maps ϕthat are self-maps of the disk and then review the transformation ϕ7→ ϕ∗between linear fractional maps used by Cowen [1], [2] and also by other authors [3], [5]. In the second part of the paper, we use a new reasoning with the orthogonal basis of the quotient Dirichlet space to obtain the formula for the adjoint of a composition operator with linear fractional symbol C∗ ϕwhen Cϕacts in D0. Acknowledgments The contents of this note constitute part of the author’s thesis at the Universidad Aut´onoma de Madrid under the supervision of Professor D. Vukoti´c. The author would like to use this opportunity to thank her advisor for his encouragement and help. 1. Preliminaries 1.1. Pseudo-hyperbolic disks. Denote by ϕathe disk automorphism which is an involution and interchanges the points 0 and a: (3) ϕa(z) = a−z 1−az , a ∈D. The pseudo-hyperbolic disk of center aand radius ris defined as ∆(a, r) = {z∈D:|ϕa(z)|< r}(0 < r < 1) . Observe that it is a Euclidean disk since (4) ∆(a, r) = ϕa(D(0, r)). Conversely, every Euclidean disk D(c, R) = {z∈C:|z−c|< R < 1}is a pseudo-hyperbolic disk whose pseudo-hyperbolic radius rand center acan be computed according to the following formulas: r=1 + R2− |c|2−p(1 + R2− |c|2)2−4R2 2R, a=2|c| 1−R2+|c|2+p(1 −R2+|c|2)2−4R2. COMPOSITION OPERATORS WITH LINEAR FRACTIONAL SYMBOLS 107 See Lemma 4 of [9]. 1.2. Linear fractional self-maps of the disk. We say that ϕis a linear fractional map if it has the form ϕ(z) = az+b cz+dfor complex numbers a, b, c, d such that ad −bc 6= 0. We now present some properties of linear fractional maps that we will use in Section 3. The following important lemma was first proved in [1]. Lemma 1.Let ϕ(z) = az+b cz+dbe a linear fractional map. Then ϕis a self-map of the disk if and only if the linear fractional transformation (5) ϕ∗(z) = 1 ϕ−1(1 z)=az −c −bz +d is also a self-map of the disk. Proof. Denote by Cthe extended complex plane. To prove that ϕ∗is a selfmap of the disk, we just need to observe that ϕis a self-map of the disk itself and this implies that ϕ−1(C\D)⊂C\D. On the other hand, the map 1/z is a one-to-one transformation from Donto C\D. These two facts give us the desired conclusion. The second identity in (5) follows by a direct calculation. ¤ Lemma 2.Let ϕand ψbe two linear fractional self-maps of D. Then (ϕ◦ψ)∗=ψ∗◦ϕ∗. Proof. Straightforward. ¤ Once we know ϕis a linear fractional self-map of the disk, Lemma 1 shows us how to build a related linear fractional transformation ϕ∗from the disk into itself which is useful in the study of composition operators. However, this does not help us in deciding whether ϕactually maps Dinto itself. We will do this in the next section. 2. A characterization of the linear fractional self-maps of the disk The following theorem tells us when a linear fractional map is a self-map of the unit disk only in terms of its coefficients a, b, c, d. Due to the lack of a specific reference, we prove it here. Theorem 1.Let ϕ(z) = az+b cz+dbe a linear fractional map. Then ϕ(D)⊂Dif and only if (6) |bd −ac|+|ad −bc|≤|d|2− |c|2. 108 M. J. MART´ IN Proof. First of all, we observe that if ϕis a self-map of the disk, then d6= 0 and | − d/c|has to be greater than or equal to 1 whenever c6= 0. It is known that every univalent self-map of the disk ϕsuch that ϕ(D) is a Euclidean disk of center A∈Dand radius Rcan be written in the form ϕ=λRϕα+A, |A|+R≤1,|λ|= 1 and ϕαas in (3): just observe that ψ= (ϕ−A)/R is a disk automorphism. We now have to solve the equation (7) az +b cz +d=λRϕα(z) + A. Recall that ad −bc 6= 0. Without loss of generality, we may assume that d= 1. So (7) becomes: az +b cz + 1 =−(Aα +λR)z+ (λRα +A) −αz + 1 . This is only possible when the corresponding coefficients are equal: a=−(Aα +λR), b =λRα +A, c =−α. This yields R=Ac −a λ, A =b−ac 1− |c|2, hence R=bc −a λ(1 − |c|2)=|bc −a| 1− |c|2. Returning to the general case when dis not necessarily 1, we get (8) R=|bc −ad| |d|2− |c|2. Similarly, in the general case, (9) A=bd −ca |d|2− |c|2. So, ϕwill be a self-map of the disk if and only if |A|+R≤1 or, equivalently, |bd −ac|+|ad −bc| ≤ |d|2− |c|2. ¤ We remark that in 1917 Schur gave a criterion in terms of the Taylor coefficients for a general analytic function to be a self-map of the disk. See [4] or [7] for an elegant proof. However, even for linear fractional maps this criterion does not seem easy to use. It is not clear how it would imply our Theorem 1. COMPOSITION OPERATORS WITH LINEAR FRACTIONAL SYMBOLS 109 Observe that using both Theorem 1 and Lemma 1, we have the following curious fact about complex numbers. Corollary 1.Let a, b, c, d be complex numbers such that ad −cb 6= 0. Then |bd −ac|+|ad −bc| ≤ |d|2− |c|2 if and only if |cd −ab|+|ad −bc| ≤ |d|2− |b|2. 3. Adjoints of composition operators with linear fractional self-maps of Das symbols One of the major problems in the study of composition operators is the lack of a reasonable representation for the adjoint C∗ ϕ. It is known that if we denote by Kωthe reproducing kernel function, the formula C∗ ϕ(Kω) = Kϕ(ω)holds in any of the spaces H2, A2,or D. Beyond this fact, not much is known about the adjoints of composition operators. We review some important results. Using the reproducing kernel functions, Cowen [1] obtained an expression for the adjoint of a composition operator with linear fractional self-map of the disk as symbol when it acts in the Hardy space H2. His result was later extended using the same method to the weighted Bergman spaces A2 αby Hurst [6]. The argument used by Cowen and Hurst is related with the particular form of the reproducing kernel functions of the Hardy and Bergman spaces and hence cannot be adapted to the Dirichlet space. However, the computation of C∗ ϕfor linear fractional symbol ϕwhen Cϕacts in the Dirichlet space has been obtained recently by Gallardo and Montes using a reasoning related to fixed points and similarity to unitary operators in certain cases [3]. In this section we present a different method in terms of the orthogonal bases of the spaces mentioned above which is available for all of them and we also prove that only a few composition operators have another composition operator as adjoint. Theorem 2.Let Cϕbe a composition operator acting in the Dirichlet space modulo constant functions, the following statements are equivalent: (i) There exists a self-map of the disk ψsuch that C∗ ϕ=Cψ. (ii) ϕ(z) = az+b cz+dis a linear fractional self-map of the disk. Moreover, if the conditions above are satisfied, ψequals the map ϕ∗given by (5). Proof. We first prove that (i) =⇒(ii). Let C∗ ϕ=Cψfor some self-maps ϕ, ψ of the disk, where ϕ(z) = P∞ n=0 anzn, ψ(z) = P∞ n=0 bnzn. For all n, m ∈N,hCψzn, zmiD0=hzn, CϕzmiD0. Taking n= 1 we get hψ, zmiD0=hz, ϕmiD0. 110 M. J. MART´ IN Keeping in mind that hAzn, BzmiD0=nAB when n=mand zero otherwise, we have for all m≥1 mbm= (ϕm)0(0) ⇐⇒ bm=ϕm−1(0)ϕ0(0) ⇐⇒ bm=am−1 0a1. Using the last equality, ψ(z) = ∞ X n=0 bnzn=b0+ ∞ X n=1 an−1 0a1zn =b0+a1z 1−a0z =(a1−a0b0)z+b0 1−a0z. Thus, ψis a linear fractional map. Repeating the process above and interchanging ψand ϕ, we have: ϕ(z) = (b1−b0a0)z+a0 1−b0z=(a1−b0a0)z+a0 1−b0z. That is, ϕis a linear fractional self-map of the disk. Comparing with equation (5), we see that ψ=ϕ∗. We now prove (ii) =⇒(i). Recall that every self-map ϕof the unit disk such that ϕ(D) is a Euclidean disk of center Aand radius Rcan be written in the form ϕ(z) = λRϕα(z) + A, where |λ|= 1 and α=ϕ−1(A). Using this fact, we will obtain the adjoint of the operator Cϕby computing the adjoints of composition operators with more elementary symbols. We first compute the adjoint of the composition operator C`where `(z) = Az+B,|A| +|B| ≤ 1. Note that, even though `is linear, `∗is not: `∗(z) = Az/(1 − Bz). We will see that hzn, C`zmiD0=hCψzn, zmiD0holds for ψ=`∗ and for all n, m ∈Nand this will prove that C∗ `=C`∗. Using the inner product given by (2), we have hzn, C`zmiD0=hzn,(Az+B)miD0 = m X j=0 m! (m−j)!j!Bm−jAjhzn, zjiD0 =   m! (m−n)!(n−1)!Bm−nAn,if m≤n; 0,otherwise. COMPOSITION OPERATORS WITH LINEAR FRACTIONAL SYMBOLS 111 On the other hand, hC`∗zn, zmiD0=*µAz 1− Bz¶n , zm+D0 =*Anzn ∞ X j=0 (j+n−1)! j!(n−1)! Bjzj, zm+D0 (10) = ∞ X k=n (k−1)! (k−n)!(n−1)!Bk−nAnhzk, zmiD0 =   m! (m−n)!(n−1)!Bm−nAn,if m≤n; 0,otherwise. =hzn, C`zmiD0. In (10) we have used the obvious identity 1 (1 − Bz)n=1 Bn(n−1)! µ1 1− Bz¶(n−1) , z ∈D. It is now left to compute C∗ ϕαwhere ϕαis an involutive automorphism as in (3). We apply the change of variable w=ϕα(z) whose Jacobian is |ϕ0 α(z)|2. Taking into account that ϕα(ϕα(w)) = w, we have for every f, g ∈ D0, hf◦ϕα, giD0=ZD f0(ϕα(z)) ·ϕ0 α(z)·g0(z)dA(z) =ZD f0(w)·g0(ϕα(w)) ·ϕ0 α(w)dA(w) =hf, g ◦ϕαiD0. It follows that C∗ ϕα=Cϕα. Finally, recalling that ϕ=`◦ϕαand that ϕ∗ α=ϕα, we obtain C∗ ϕ=C∗ `◦ϕα= (CϕαC`)∗=C`∗Cϕα=Cϕα◦`∗=Cϕ∗=Cψ, using also Lemma 2. ¤ By applying the same method, we obtain the following result known to the experts, but not easy to find explicitly stated in the literature. Note that the statement now refers to the true Dirichlet space instead of D0. Proposition 1.Let Cϕbe a composition operator acting in the true Dirichlet space D. Then C∗ ϕ=Cψfor some self-map of the disk ψif and only if ϕ(z) = az, a ∈D. In this case, ψ(z) = ϕ∗(z) = az. Proof. The sufficiency of the condition ϕ(z) = az follows easily from the equality hanzn, zmiD=hzn, anzmiD, n, m ∈N∪ {0}. 112 M. J. MART´ IN To prove the necessity, it is enough to repeat the process in the proof of (i)=⇒(ii) of Theorem 2 and to observe that for every n, m ∈N, hCψzn, zmiD=Cψzn(0) ·0m+hCψzn, zmiD0. So, whenever m6= 0, hCψzn, zmiD=hCψzn, zmiD0. On the other hand, for m= 0, taking n= 1, by the definition of the inner product in Dwe obtain hCψz, 1iD=hz, Cϕ1iD=hz, 1iD=⇒ψ(0) = b0= 0, hCϕz, 1iD=hz, Cψ1iD=hz, 1iD=⇒ϕ(0) = a0= 0. Thus, we have ϕ(z) = a1zand ψ(z) = a1z.¤ References [1] Cowen, C. C. Linear fractional composition operators on H2, Integral Equations Operator Theory 11 (1988), 151–160. [2] Cowen, C.; MacCluer, B. Composition Operators on Spaces of Analytic Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton 1995. [3] Gallardo-Guti´errez, E.; Montes-Rodr´ıguez, A. Adjoints of linear fractional composition operators on the Dirichlet space, Math. Ann. 327 (2003), 117–134. [4] Garnett, J. Bounded Analytic Functions, Academic Press, New York 1981. [5] Hammond, Ch. On the norm of a composition operator with linear fractional symbol, Acta Sci. Math. (Szeged) 69 (2003), 813–829. [6] Hurst, P. R. Relating composition operators on different weighted Hardy spaces, Arch. Math. 68 (1997), 503–513. [7] Kortram, R. A. A simple proof of Schur’s Theorem, Proc. Amer. Math. Soc. 129 (1931), 3211–3212. [8] Shapiro, J. H. Composition Operators and Classical Function Theory, Springer-Verlag, New York 1993. [9] Vukoti´c, D. On norms of composition operators acting on Bergman spaces, J. Math. Anal. Appl. 291 (2004), 189–202. (Erratum to appear). Departamento de Econom´ ıa, Universidad Carlos III, C/ Madrid 126, 28903 Getafe (Madrid), Spain E-mail address:[email protected]