Composition Operators on the Bloch Space of Several Complex Variables

Let ϕ and ψ be holomorphic self-maps of the unit disk, and denote by C ϕ , C ψ the induced composition operators. This paper gives some simple estimates of the essential norm for the difference of composition operators C ϕ −C ψ from Bloch spaces to Bloch spaces in the unit disk. Compactness of the difference is also characterized.

In this paper, we characterise the analytic functions ϕ mapping the open unit disk ▵ into itself whose induced composition operator Cϕ: f ↦ f ∘ ϕ is an isometry on the Bloch space. We show that such functions are either rotations of the identity function or have a factorisation ϕ = gB where g is a non-vanishing analytic function from Δ into the closure of ▵, and B is an infinite Blaschke product whose zeros form a sequence{zn} containing 0 and a subsequence satisfying the conditions , and

The paper contains a new proof for the sufficiency in Joel H. Shapiro’s recent characterization of inner functions saying that an analytic self-map φ of the open unit disk is an inner function if and only if the essential norm of the composition operator of symbol φ is equal to √ (1 + |φ(0)|)/(1 − |φ(0)|). The main ingredient in the proof is a formula for the essential norm of a composition operator in terms of Aleksandrov measures obtained by Cima and Matheson. The necessity was originally proved by Joel Shapiro in 1987. A short proof of the necessity, by Aleksandrov measure techniques, was obtained by Jonathan E. Shapiro in 1998. For each holomorphic self-map φ : U → U of the open unit disk U, the composition operator Cφ of symbol φ is defined as follows Cφf = f ◦ φ, f ∈ H. In this definition H is the Hilbert Hardy space on U, i.e., the set of all analytic functions on U with square summable Taylor coefficients. It is well known that each such φ induces a bounded composition operator Cφ on H. Recently Joel Shapiro obtained the following characterization of inner functions [7], (that is of self-maps φ whose radial limit function is unimodular almost everywhere on the unit circle T). Theorem 1. The function φ : U → U is inner if and only if (1) ‖Cφ‖e = √ 1 + |φ(0)| 1− |φ(0)| , where ‖Cφ‖e denotes the essential norm of Cφ. The necessity in the previous equivalence was originally proved in [6]. Jonathan Shapiro [5] observed that Aleksandrov measures can be used Received by the editors on May 7, 2003, and in revised form on May 30, 2003. Copyright c ©2005 Rocky Mountain Mathematics Consortium 1723

Let Ω1, Ω2be open subsets of ℝand ℝ, respectively, and let A(Ω1) denote the space of real analytic functions on Ω1. We prove a Glaeser type theorem by characterizing when a composition operatorCφ: A(Ω1) → A(Ω2),Cφ(f) ≔f∘φ, is a topological embedding. Using this result we characterize when A(Ω1) can be embedded topologically into A(Ω2) as a locally convex space or as a topological algebra. We also characterize LB–subspaces and Fréchet subspaces of A(Ω1). In particular, it follows that if A(Ω1) and A(Ω2) are isomorphic as locally convex spaces, thend1=d2.

We provide several characterizations of the bounded and the compact weighted composition operators from the Bloch space and the analytic Besov spaces (with ) into the Zygmund space . As a special case, we show that the bounded (resp., compact) composition operators from , , and to coincide. In addition, the boundedness and the compactness of the composition operator can be characterized in terms of the boundedness (resp., convergence to 0, under the boundedness assumption of the operator) of the Zygmund norm of the powers of the symbol.

We give a new and simple compactness criterion for composition operators C ϕ on BMOA and the Bloch space in terms of the norms of ϕ n in the respective spaces.

While there is a large variety of univalently induced closed range composition operators on the Bloch space, we show that the only univalently induced, closed range, composition operators on the Bloch-type spaces Bα with α ≠ 1 are the ones induced by a disc automorphism.

On the Bergman space in the unit disk, we study the intertwining relation for Volterra type operators, whose intertwining operator is a composition operator. We also investigate the “compact” intertwining relations for Volterra type operators. As obvious consequences, the essential commutativity of Volterra type and composition operators are characterized. At the end of the paper, we find a new connection between the Bergman space and little Bloch space through this essential commutativity.

We call a sequence {zn} in the open unit disk Δalmost thin if \({\mathop {\lim \sup }\limits_{n \to \infty } }\Pi {\left( {z_{n} } \right)} = 1,\) where Π(zn) is the product of the pseudo-hyperbolic distances between zn and the other elements in the sequence. We call the sequence thick if \({\mathop {\lim }\limits_{n \to \infty } }\Pi {\left( {z_{n} } \right)} = 0.\) In this paper, we use elementary complex variables methods to prove a sharper form of a characterization of the symbols of the isometric composition operators on the Bloch space proved by Martin and Vukotic in terms of the preimages of one-point sets. Specifically, we show that a bounded analytic function φ from Δ into itself has Bloch semi-norm equal to one (which is the largest possible) if and only if it is a conformal automorphism of Δ or if the preimage of each point of Δ contains a sequence along which the hyperbolic derivative of φ approaches one (condition (M’)). In particular, the preimage of each point of Δ is an almostthin sequence. Using similar techniques we also show that a bounded analytic function f is in the little Bloch space if and only if for each \(a\in \mathbb{C}\) such that \(f^{-1}(a)=\{z_{n}:n\in \mathbb{N}\}\) is infinite $$ {\mathop {\lim }\limits_{n \to \infty } }\left(1 - |z_{n} |^{2} \right)|f'{\left( {z_{n} } \right)}| = 0. $$ Consequently, an infinite Blaschke product whose left composition by a conformal automorphism of Δ is always a Blaschke product belongs to the little Bloch space if and only if the preimage of every point in Δ is a thick sequence. We give an example to show that this characterization does not extend to unbounded Bloch functions. We also prove an analogous characterization of the little normal functions. Finally, we give a sufficient condition on a holomorphic self-map of the polydisk Δn to be the symbol of an isometric composition operator on the Bloch space of Δn and discuss a higher-dimensional analogue of condition (M’).

We characterize all isometries among the composition operators acting on the Bloch space in terms of the hyperbolic derivative and cluster set of the symbol, and display a class of nontrivial examples.

Hypercyclic composition operators on the little Bloch space and the Besov spaces

Let be the unit disk in the complex plane. We define to be the little Bloch space of functions f analytic in which satisfy lim|z|→1(1 - |z|2)|f'(z)| = 0. If is analytic then the composition operator C φ : f ↦ f ∘ φ is a continuous operator that maps into itself. In this paper, we show that the compactness of C φ , as an operator on , can be modelled geometrically by its principal eigenfunction. In particular, under certain necessary conditions, we relate the compactness of C φ to the geometry of , where σ satisfies Schöder's functional equation σ ∘ φ = φ'(0)σ.2000 Mathematics Subject Classification: Primary 30D05; 47B33 Secondary 30D45.

We study the weighted composition operators Wh,φ on Hardy space \(H^2 (\mathbb{B})\) whenever h ∈ BMOA (resp. h ∈ VMOA). Analogous results are given for \(H^p (\mathbb{B})\) spaces and the scale of weighted Bergman spaces. In the latter case, BMOA is replaced by the Bloch space (resp. VMOA by the little Bloch space).

In this paper, we consider the composition operators on harmonic Bloch-type spaces. Then first we compute their spectra on harmonic α-Bloch spaces and harmonic little α-Bloch space and then we characterize isometric composition operators on harmonic α-Bloch-type spaces. Indeed we obtain a relation between the properties of the inducing function φ and the isometricity of the composition operator Cφ.

In this paper, function spaces Q(p)(B) and Q(p,o)(B), associated with the Green's function, are defined and studied for the unit ball B of C-n. We prove that Q(p)(B) and Q(p,o)(B) are Mobius invariant Banach spaces and that Q(p)(B) = Bloch(B), Q(p,o)(B) = B-o(B) (the little Bloch space) when 1 < p < n/(n - 1),Q(1) = BMOA(partial derivative B) and Q(1,0)(B) = VMOA(partial derivative B). This fact makes it possible for us to deal with BMOA and Bloch space in the same way. And we give necessary and sufficient conditions on boundedness (and compactness) of the Hankel operator with antiholomorphic symbols relative to Q(p)(B) (and Q(p,o)(B)). Moreover, other properties about the above spaces and \phi(z)(omega)\,phi(z)(omega) is an element of Aut(B), are obtained.