Composition operators on holomorphic Sobolev spaces in [formula omitted

COMPOSITION OPERATORS ON HARDY-ORLICZ SPACES

(Communicated by G. Sinnamon) Abstract. Suppose ϕ is an analytic self-map of open unit disk D and w is an analytic func- tion on D. Then a weighted composition operator induced by ϕ with weight w is given by (Ww,ϕ f)(z )= w(z)f(ϕ(z)) ,f orz ∈ D and f analytic on D.W efind a sufficient condition un- der which two composition operators lie in the same path component of C(H 2 ) ,a nd wefind as ufficient condition for the difference of such operators to be compact on H 2 (D) .T hen we provide another example that answers a question raised by Shapiro and Sundberg (18 )n ega- tively. Moreover, we characterize the Hilbert-Schmidt difference of two composition operators on H2(D).

Let $\varphi,\psi$ be the analytic self-maps of the unit ball ${\mathbb B}$, we characterize the Hilbert-Schmidt differences of two composition operator $C_\varphi$ and $C_\psi$ on weighted Bergman space $A_\alpha^2$, and give some conclusions about the topological structure of $\mathcal{C}(A_\alpha^2)$, the space of all bounded composition operators on $A_\alpha^2$ endowed with operator norm.

Suppose $\\varphi$ is an analytic self-map of open unit disk $\\mathbf{D}$ and $\\psi$ is an analytic function on $\\mathbf{D}$. Then a weighted composition operator induced by $\\varphi$ with weight $\\psi$ is given by $(W_{\\psi, \\varphi}f)(z)= \\psi(z)f(\\varphi(z))$, for $z \\in \\mathbf{D}$ and $f$ analytic on $\\mathbf{D}$. Necessary and sufficient conditions are given for the boundedness and compactness of the weighted composition operators $W_{\\psi, \\varphi}$. In terms of fixed points in the closed unit disk $\\overline{\\mathbf{D}}$, conditions under which $W_{\\psi, \\varphi}$ is compact are given. Necessary conditions for the compactness of $C_{\\varphi}$ are given in terms of the angular derivative $\\varphi^{\\prime}(\\zeta)$ where $\\zeta$ is on the boundary of the unit disk. Moreover, we present sufficient conditions for the membership of composition operators in the Schatten $p$-class $S_{p}(H^s(\\beta_1), H^q(\\beta_2))$, where the inducing map has supremum norm strictly smaller than~$1$.

It is shown that there exist analytic self-maps ϕ of the unit disc inducing compact composition operators on the Hardy space , 1 ≤ p < ∞ such that the Hausdorff dimension of the set is one; sharpening a classical result due to Schwartz. Moreover, the same holds in the weighted Dirichlet spaces with 0 < α < 1. As a consequence, we deduce that there exist symbols ϕ inducing compact composition operators on such that the α-capacity of E ϕ is positive, which is no longer true for those just inducing Hilbert-Schmidt composition operators on .

In this paper we study bounded and compact weighted composition operator, induced by a fixed analytic function and an analytic self-map of the open unit disk, from Bergman space into weighted Bloch space. As a corollary, obtain the characterization of composition operator from Bergman space into weighted Bloch space.

For a wide class of radial weights we calculate the essential norm of a weighted composition operator \({uC_\varphi}\) on the weighted Banach spaces of analytic functions in terms of the analytic function \({u \colon \mathbb D \to \mathbb C}\) and the nth power of the analytic selfmap \({\varphi}\) of the open unit disc \({\mathbb D}\) . We also apply our result to calculate the essential norm of composition operators acting on Bloch type spaces with general radial weights.

In this work, we consider refined geometric characterizations of mappings that generate bounded $(p,q)$-composition operators on Sobolev spaces. In the case $n-1&lt;q\leq p&lt;\infty$, geometric characterizations in terms of metric and measure distortions are given with detailed proofs.

Let \({u \in \mathcal{H}(\mathbb{D})}\) and φ be an analytic self-map of \({\mathbb{D}}\). We estimate the essential norms of weighted composition operators uCφ acting on Zygmund type spaces in terms of u, φ, their derivatives and the n-th power φn of φ. Moreover, we give similar characterizations for boundedness of uCφ between Zygmund type spaces.

Let u be an analytic function in the unit disk mathbb{D} and φ be an analytic self-map of mathbb{D}. We give characterizations of the symbols u and φ for which the multiplication operator M_{u} and the weighted composition operator M_{u,varphi } are isometries of BMOA.

Let ϕ be an analytic self-map of the open unit disk D. Such a map induces a composition operator Cϕ acting on weighted Bergman spaces of infinite order. We study when such operators are power bounded and when they are uniformly mean ergodic.

Let E be a complex Banach space with open unit ball $$B_E.$$ For analytic self-maps $$\varphi $$ of $$B_E$$ with $$\varphi (0) =0,$$ we investigate the spectra of weighted composition operators $$uC_\varphi $$ acting on a large class of spaces of analytic functions. This class contains, for example, weighted Banach spaces of $$H^\infty $$-type on $$B_E$$, weighted Bergman spaces $$A^p_\alpha ({\mathbb {B}}_N)$$ and Hardy spaces $$H^p({\mathbb {B}}_N).$$ We present a general approach for deducing new information about the spectrum and for estimating the essential spectral radius of $$uC_\varphi .$$

Let φ and ψ be analytic self-maps of the open unit disk D . Using pseudo-hyperbolic distance ρ(φ ,ψ) , we characterize the boundedness and compactness of the differences of generalized composition operators (C φ −Ch ψ ) f (z) = ∫ z 0 [ f ′(φ(ξ ))g(ξ )− f ′(ψ(ξ ))h(ξ )]dξ , z ∈ D between two Bloch-type spaces on D . The results generalize the corresponding results on the single generalized composition operator and on the differences of generalized composition operators on the Bloch space. Mathematics subject classification (2010): Primary 47B38; secondary 30H30.

Let0&lt;p&lt;∞, let-2&lt;q&lt;∞, and letφbe an analytic self-map of𝔻andg∈H(𝔻). The boundedness and compactness of generalized composition operators(Cφgf)(z)=∫0z‍f'(φ(ξ))g(ξ)dξ, z∈𝔻, f∈H(𝔻), fromℬμ(ℬμ,0) spaces toQK,ω(p,q)spaces are investigated.

Let ψ be an analytic map and ϕ an analytic self-map of the open unit disk 𝔻 {\mathbb{D}} . Such maps induce the so-called weighted composition operator C ϕ , ψ : H ⁢ ( 𝔻 ) → H ⁢ ( 𝔻 ) {C_{\phi,\psi}:H(\mathbb{D})\to H(\mathbb{D})} , f ↦ ψ ⁢ ( f ∘ ϕ ) {f\mapsto\psi(f\circ\phi)} , where H ⁢ ( 𝔻 ) {H(\mathbb{D})} denotes the set of all analytic functions on 𝔻 {\mathbb{D}} . We study when such an operator acting between different weighted Bergman spaces has a closed range.