Cyclic behavior of linear fractional composition operators

E extracta mathematicae Vol. 16, N´um. 1, 147 – 152 (2001) Cyclic Behavior of Linear Fractional Composition Operators E. A. Gallardo-Guti´ errez, A. Montes-Rodr´ ıguez Dpto. de Matem´aticas, Univ. de C´adiz, Apartado 40, 11510-Puerto Real, C´adiz, Spain Dpto. de An´alisis Matem´atico, Univ. de Sevilla, Avda. Reina Mercedes, Apartado 1160, 41080-Sevilla, Spain e-mail: [email protected], [email protected] (Research announcement presented by M. Gonz´alez) AMS Subject Class. (2000): 47B38, 30D05 Received October 20, 2000 1. Cyclic linear fractional composition operators For each real number νthe weighted Dirichlet space Sνis the space of analytic functions f(z) = P∞ n=0 anznsuch that the following norm kfk2 ν= ∞ X n=0 |an|2(n+ 1)2ν is finite. Observe that k · kνcomes from an inner product and, therefore, the spaces Sνare Hilbert spaces. Also, notice that the above condition on the Taylor coefficients of the function fimplies that lim supn→∞ |an|1/n ≤1 and, thus, each f∈ Sνis analytic, at least, on the unit disk Dof the complex plane. For some values of νthe spaces Sνare very well known classical analytic function spaces: for ν= 1/2 it is the Dirichlet space D; for ν= 0 it is the Hardy space H2and for ν=−1/2 it is the Bergman space A2. The easiest composition operators are those induced by linear fractional maps ϕ(z) = az +b cz +dad −bc 6= 0 and ϕ(D)⊂D. In this case, the linear fractional composition operator defined by Cϕf=f◦ϕ(f∈ Sν) 147 148 e.a. gallardo-guti´ errez, a. montes-rodr´ ıguez is always bounded on any of the Sνspaces (see [8], for instance). In some of the Sνspaces, this fact fails to be true for certain holomorphic selfmaps of D(see [4]). As weighted shifts, composition operators are a class of concrete operators that, in general, do not fall into a class of operators satisfying a prescribed property invariant under similarity. This along several nice features makes them worth studying. A bounded linear operator Tacting on a separable Hilbert space His said to be cyclic if there is an f∈ H such that the linear span of the orbit {Tnf}n≥0is dense in H. There have been much interest for a long time in cyclic operators because their relation to invariant subspaces. More recently, stronger forms of cyclicity are being investigated. If there is f∈ H such that {λTnf:λ∈Cand n= 0,1, . . .}is dense in H, then the operator Tas well as the vector fare called supercyclic. If the orbit itself is dense in H, then T and fare called hypercyclic. In [5] the authors have completely characterized the cyclicity and hypercyclicity of scalar multiples of linear fractional composition operators in all the Sνspaces (see Table I and [2] for the classification of linear fractional maps). Previous results for ν= 0 and ν > 1/2 (or ν > 3/2, depending on which type of ϕ) were already known (see [3] and the page 150). Cyclic composition operators for ν= 0 were first studied in [13]. Observe the central role played by the Dirichlet space Din the cut-off of any of the cyclic properties. Since supercyclicity is an intermediate property between cyclicity and hypercyclicity, many of the results for the supercyclicity column follow immediately. If ϕhas a fixed point in D(as in the elliptic case), it is easy to show that there is no supercyclic operators (see [5, Sect. 8]). For the Dirichlet space D(ν= 1/2), the supercyclicity follows because λCϕis hypercyclic whenever |λ|>1 (see [5, Sect. 4]). Therefore, to complete the characterization it is just sufficient to know the supercyclic behavior of composition operators induced by parabolic non automorphisms. In this sense, Shapiro [11], using that Cϕis similar to λCϕacting on certain subspace, had proved that λCϕis not hypercyclic on H2. But this is far to mean that Cϕis supercyclic on H2. We stress here that even an invertible operator can be supercyclic and not hypercyclic for any scalar multiple of it. This can be seen as a consequence of Theorem 5.2 in [7]. Better still, Salas [10] has provided examples of this behavior in which even the set of normal eigenvalues is the empty set. The following result that completes the supercyclic behavior of linear fractional composition operators in the Hardy space H2was firstly proved in [6]. supercyclic operators 149 Theorem 1. Let ϕbe a parabolic non automorphism that takes the unit disk into itself. Then Cϕacting on the Hardy space H2is not supercyclic. Sketch of the proof. Parabolic non automorphisms that take the unit disk into itself have just one fixed point on the boundary ∂Dand because any cyclic property is invariant under similarity it may be assumed that ϕ(1) = 1 and, thus, the following formula holds ϕ(z) = (2 −a)z+a −az +2+a where <a > 0 because ϕis not an automorphism of D. The method of the proof is to get estimates from above and from below of Cϕacting on certain subspaces. These subspaces are built up from the eigenfunctions of Cϕ. For each t≥0 an elementary computation shows that the inner function et(z) = exp ·tz+ 1 z−1¸ is an eigenfunction of Cϕin H2corresponding to the eigenvalue e−at (see [4] for more about the spectrum of Cϕ). The proof also uses the following result span {et:t≥0}=H2 that is already contained in Ahern and Clark’s work [1]. By taking quotients by the Cϕ-invariant subspace F= span {eτ, eσ}, where τ > σ > 0, the space b H=H2/F can be decomposed in an orthogonal sum with some useful properties b H2=b Xτ⊕b Zτσ ⊕b Yσ.(1) First, if σis chosen large enough, then there are constants 0 < c < C such that the operator b Cϕdefined by b Cϕˆ f=d Cϕfsatisfies kb Cn ϕ¯¯ b Yσ ˆ fk ≤ cnkˆ fkfor ˆ f∈b Yσand kb Cn ϕ¯¯ b Xτ ˆ fk ≥ Cnkˆ fkfor ˆ f∈b Xτ. Second, the spaces in (1) are b Cϕ-invariant. The estimates above can be obtained by using Gerschgorin’s Theorem (see [12]) about localization of eigenvalues. This theorem is applied to n×nmatrices that represent the restrictions of b Cϕto certain n-dimensional b Cϕ-invariant subspaces. The matrices are obtained thank to very nice orthogonality properties that possess the eigenfunctions et(z) in the Hardy space H2. 150 e.a. gallardo-guti´ errez, a. montes-rodr´ ıguez Finally, suppose that Cϕis supercyclic, then so is b Cϕ. Now, choose a supercyclic vector b f=ˆ fτ⊕ˆ fτσ ⊕b fσ. Then ˆ fτmust be different from zero; otherwise b Cϕis not supercyclic. Finally, for ˆg6= 0 orthogonal to b Xτ, we have |h b Cn ϕˆ f, ˆgi| kb Cn ϕˆ fkkˆgk≤cnkfσk Cnkfτk that goes to zero. Therefore, Cϕis not supercyclic, a contradiction. The last idea in the sketch of the proof has occurred first in [9] in relation to infinite dimensional subspaces of supercyclic vectors (see [6] for more details). Type of ϕCyclic Supercyclic Hypercyclic Example Hyperbolic Automorphism ν < 1/2ν < 1/2ν < 1/23z+ 1 z+ 3 Parabolic Automorphism ν < 1/2ν < 1/2ν < 1/2(1 + i)z−1 z+i−1 Hyperbolic Non- Automorphism Always ν≤1/2ν < 1/21 + z 2 Parabolic Non- Automorphism ν≤3/2 Never Never 1 2−z Interior & Exterior Always Never Never −z 2 + z Interior & Boundary Never Never Never z 2−z Elliptic Irrational Rotation Always Never Never e2i/3z Elliptic Rational Rotation Never Never Never e2πi/3z Table 1 supercyclic operators 151 The comparison principle for supercyclic operators (first noticed in [10]) only shows that Cϕis not supercyclic for any Sνfor ν≥0. However, a careful analysis shows that the methods we have just described can also be used to prove that Cϕis not supercyclic in any of the Sνspaces. Although there is no good orthogonality properties of the functions et(z) in Sνfor ν > 0, one can still obtain rather manageable Cϕ-invariant subspaces. In fact, we can prove the following result that completes the cyclic behavior of linear fractional composition operators in all the Sνspaces. Theorem 2. Let ϕbe a parabolic non automorphism that takes the unit disk into itself. Then Cϕis not supercyclic in any of the weighted Dirichlet spaces Sν. Actually, our methods are still valid for spaces of analytic functions that are not Hilbert spaces, and can be used to get non supercyclicity of Cϕ, for instance, in the Bergman spaces Ap, 1 ≤p < ∞(and by the comparison principle in any space that is densely contained in some of these spaces and has a stronger topology). The comparison principle is one of the motivations for the work discussed in this note (see [5] for more details). References [1] Ahern, P.R., Clark, D.N., On functions orthogonal to invariant subspaces, Acta Mathematica, 124 (1970), 191 – 204. [2] Ahlfors, L.V., “Complex Analysis”, McGraw-Hill, New York, 1979. [3] Bourdon, P.S., Shapiro, J.H., Cyclic composition operators on H2, Proc. Symp. Pure Math., 51 (2) (1990), 43 – 53. [4] Cowen, C., MacCluer, B., “Composition Operators on Spaces of Analytic Functions”, CRC Press, 1995. [5] Gallardo-Guti´ errez, E.A., Montes-Rodr´ ıguez, A., The role of the spectrum in the cyclic behavior of composition operators, preprint. [6] Gallardo-Guti´ errez, E.A., Montes-Rodr´ ıguez, A., The role of the angle in the supercyclic behavior, preprint. [7] Gonz´ alez, M., Le´ on-Saavedra, F., Montes-Rodr´ ıguez, A., Semi-Fredholm theory: hypercyclic and supercyclic subspaces, Proc. London Math. Soc., 81 (3) (2000), 169 – 189. [8] Hurst, P.R., Relating composition operators on different weighted Hardy spaces, Arch. Math., 68 (1997), 503 – 513. [9] Montes-Rodr´ ıguez, A., Salas, H.N., Supercyclic subspaces: spectral theory and weighted shifts, preprint. [10] Salas, H.N., Supercyclicity and weighted shifts, Studia Math., 135 (1999), 55 – 74. [11] Shapiro, J.H., “Series of Lectures in the Hypercyclicity/Subnormal Seminar”, Michigan State University, April, 1998. 152 e.a. gallardo-guti´ errez, a. montes-rodr´ ıguez [12] Zalewska-Mitura, A., Zem´ anek, J., The Gerschgorin discs under unitary similarity orbits, Linear operators. Banach Center Publications, (1997), 427 – 441. [13] Zorboska, N., “Composition Operators on Weighted Hardy Spaces”, Thesis, University of Toronto, 1987. [14] Zorboska, N., Cyclic composition operators on smooth weighted Hardy spaces, Rocky Mountain Journal of Mathematics, 29 (1999) 725 – 740.