Isometric composition operators on the weighted Hardy spaces

The notion of a completely hyperexpansive operator has been introduced in [2] by Athavale. In this paper bounded and unbounded hyperexpansive composition operators are investigated. A unique semispectral measure is associated with a bounded completely hyperexpansive composition operator. Examples of bounded and unbounded densely defined $2$-isometric (completely hyperexpansive, $2$-hyperexpansive) composition operators with invariant domains are given.

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~$1$.

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.

We propose a method for designing accurate interpolation formulas on the real axis for the purpose of function approximation in weighted Hardy spaces. In particular, we consider the Hardy space of functions that are analytic in a strip region around the real axis, being characterized by a weight function $w$ that determines the decay rate of its elements in the neighborhood of infinity. Such a space is considered as a set of functions that are transformed by variable transformations that realize a certain decay rate at infinity. Popular examples of such transformations are given by the single exponential (SE) and double exponential (DE) transformations for the SE-Sinc and DE-Sinc formulas, which are very accurate owing to the accuracy of sinc interpolation in the weighted Hardy spaces with single and double exponential weights $w$, respectively. However, it is not guaranteed that the sinc formulas are optimal in weighted Hardy spaces, although Sugihara has demonstrated that they are near optimal. An explicit form for an optimal approximation formula has only been given in weighted Hardy spaces with SE weights of a certain type. In general cases, explicit forms for optimal formulas have not been provided so far. We adopt a potential theoretic approach to obtain almost optimal formulas in weighted Hardy spaces in the case of general weight functions $w$. We formulate the problem of designing an optimal formula in each space as an optimization problem written in terms of a Green potential with an external field. By solving the optimization problem numerically, we obtain an almost optimal formula in each space. Furthermore, some numerical results demonstrate the validity of this method. In particular, for the case of a DE weight, the formula designed by our method outperforms the DE-Sinc formula.

Let X be a set of analytic functions on the open unit disk D, and let phi be an analytic function on D such that phi(D) is contained in D and f |-> f o phi takes X into itself. We present conditions on X ensuring that if f |-> f o phi is invertible on X, then phi is an automorphism of D, and we derive a similar result for mappings of the form f |-> psi.(f o phi), where psi is some analytic function on D. We obtain as corollaries of this purely function-theoretic work, new results concerning invertibility of composition operators and weighted composition operators on Banach spaces of analytic functions such as S^p and the weighted Hardy spaces H^2(beta).

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

In this paper, we discuss the isometries of composition operators on the holomorphic general family function spaces F(p, q, s). First, we classify the isometric composition operators acting on a general Banach spaces. For 1 < p < 2, we display that an isometry of Cφ is caused only by a rotation of the disk. We scrutinize the previous work on the case for p ≥ 2. Also, we characterize many of the foregoing results about all α-Besov-type spaces F(p, αp − 2, s), α > 0. We exhibit that in every classes F(p, αp − 2, s) except for the Dirichlet space D = F(2, 0, 0), rotations are the only that produce isometries.

Essential norm of composition operators between weighted Hardy spaces

The present paper introduces the class of weighted generalized composition operators of higher order defined on the weighted Hardy spaces. The study of bounded operators belonging to this class is undertaken and an attempt is made to describe their structural and spectral properties.

The classical Hardy space H2 can be viewed as the closure of the analytic polynomials in L2 of the unit circle. One generalization of H2 can be obtained by taking the closure of the analytic polynomials in £2(μ), where μ is a positive Borel measure on the unit circle. This space, a weighted Hardy space, is denoted by Η2(μ).

Complex symmetry of invertible composition operators

Let X be a ball quasi-Banach function space satisfying some mild assumptions and H X (R n ) the Hardy space associated with X. In this article, the authors introduce both the Hardy space H X (R n+1 + ) of harmonic functions and the Hardy space H X (R n+1 + ) of harmonic vectors, associated with X, and then establish the isomorphisms among , where H X,2 (R n+1 + ) and H X,2 (R n+1 + ) are, respectively, certain subspaces of H X (R n+1 + ) and H X (R n+1 + ). Using these isomorphisms, the authors establish the first order Riesz transform characterization of H X (R n ). The higher order Riesz transform characterization of H X (R n ) is also obtained. The results obtained in this article have a wide range of generality and can be applied to the classical Hardy space, the weighted Hardy space, the Herz-Hardy space, the Lorentz-Hardy space, the variable Hardy space, the mixed-norm Hardy space, the local generalized Herz-Hardy space, and the mixed-norm Herz-Hardy space.

The topological structure of the set of (weighted) composition operators has been studied on various function spaces on the unit disc such as Hardy spaces, the space of bounded holomorphic functions, weighted Banach spaces of holomorphic functions with sup-norm, Hilbert Bergman spaces. In this paper we consider this problem for all Bergman spaces $$A_{\alpha }^p$$ with $$p \in (0, \infty )$$ and $$ \alpha \in (-1, \infty )$$ . In this setting we establish a criterion for two composition operators to be linearly connected in the space of composition operators; furthermore, for the space of weighted composition operators, we prove that the set of compact weighted composition operators is path connected, but it is not a component.

We present a unified approach to the problem of characterizing the Fredholm multiplication and composition operators on the Hardy spaceH 2.

Maximal operator for multilinear Calderón–Zygmund singular integral operators on weighted Hardy spaces