Compact composition operators between weighted Bergman spaces on convex domains in ℂ n

It is shown that a compact composition operator on a weighted Bergman space over a smoothly bounded strongly convex domain in ⁿ can have no angular derivative. Also, sufficient conditions for the boundedness and the compactness of composition operators defined on Hardy and weighted Bergman spaces are obtained, for situations in which each of the target spaces is enlarged in a natural way.     

On the composition operators on the Bloch space of several complex variables

In this paper we first look upon some known results on the composition operator as bounded or compact on the Bloch-type space in polydisk and ball, and then give a sufficient and necessary condition for the composition operator to be compact on the Bloch space in a bounded symmetric domain.     

Hilbert�CSchmidt differences of composition operators on the Bergman space

In the setting of the weighted Bergman space over the unit disk, we characterize Hilbert?CSchmidt differences of two composition operators in terms of integrability condition involving pseudohyperbolic distance between the inducing functions. We also show that a linear combination of two composition operators can be Hilbert?CSchmidt, except for trivial cases, only when it is essentially a difference. We apply our results to study the topological structure of the space of all composition operators under the Hilbert?CSchmidt norm topology. We first characterize components and then provide some sufficient conditions for isolation or for non-isolation.     

Commutants of Toeplitz operators with radial symbols on the Fock–Sobolev space

In the setting of the Fock space over the complex plane, Bauer and Lee have recently characterized commutants of Toeplitz operators with radial symbols, under the assumption that symbols have at most polynomial growth at infinity. Their characterization states: If one of the symbols of two commuting Toeplitz operators is nonconstant and radial, then the other must be also radial. We extend this result to the Fock–Sobolev spaces.     

Topological structure of the space of composition operators onH ∞

Components and isolated points of the topological space of composition operators onH in the uniform operator topology are characterized. Compact differences of two composition operators are also characterized. With the aid of these results, we show that a component inC(H ) is not in general the set of all composition operators that differ from the given one by a compact operator.Supported by the Grant-in-Aid for Scientific Research (C), the Ministory of Education, Science and Culture, No. 09640218, and the Nippon Institute of Technology No. 111Supported by the Japan Society for the Promotion of Science     

Generalised weighted composition operators on Maeda–Ogasawara spaces

Every Archimedean Riesz space can be embedded as an order dense subspace of some $C^{\infty }\left(X\right)$, the Riesz space of all extended continuous functions on a Stonean space $X$, called its Maeda–Ogasawara space. Furthermore, it is a fact that every Riesz homomorphism between spaces of ordinary continuous functions on compact Hausdorff spaces is a weighted composition operator. We prove that a generalised statement holds for Maeda–Ogasawara spaces and refine these results in case the homomorphism preserves order limits.     

Weighted boundedness of some integral operators on weighted λ-central Morrey space

In this paper, the authors prove the weighted boundedness of singular integral and fractional integral with a rough kernel on the weighted λ-central Morrey space. Moreover, the weighted estimate for commutators of singular integral with a rough kernel on the weightedλ-central Morrey space is also given.     

Legendre–Fenchel transform of the spectral exponent of polynomials of weighted composition operators

For the spectral radius of weighted composition operators with positive weight e ^φ T _α , \({\varphi\in C(X)}\) , acting in the spaces L ^p (X, μ) the following variational principle holds

$\ln r(e^\varphi T_\alpha)=\max_{\nu\in M^1_\alpha} \left\{\int\limits_X\varphi d\nu-\frac{\tau_\alpha(\nu)}{p}\right\},$

where X is a Hausdorff compact space, \({\alpha:X\mapsto X}\) is a continuous mapping and τ _α some convex and lower semicontinuous functional defined on the set \({M^1_\alpha}\) of all Borel probability and α-invariant measures on X. In other words \({\frac{\tau_\alpha}{p}}\) is the Legendre– Fenchel conjugate of ln r(e ^φ T _α ). In this paper we consider the polynomials with positive coefficients of weighted composition operator of the form \({A_{\varphi, {\bf c}}= \sum_{k=0}^n e^{c_k} (e^{\varphi} T_{\alpha})^k}\) , \({{\bf c}=(c_k)\in {\Bbb R}^{n+1}}\) . We derive two formulas on the Legendre–Fenchel transform of the spectral exponent ln r(A _φ,c) considering it firstly depending on the function φ and the variable c and secondly depending only on the function φ, by fixing c.

     

Factorization of Hardy spaces and Hankel operators on convex domains in ℂ n

In this article we study the (small) Hankel operator h_b on the Hardy and Bergman spaces on a smoothly bounded convex domain of finite type in ℂⁿ. We completely characterize the Hankel operators h_b that are bounded, compact, and belong to the Schatten ideal S_p, for 0 < p < ∞. In particular, if h_b denotes the Hankel operator on the Hardy space H² (Ω), we prove that h_b is bounded if and only if b ∈ BMOA, compact if and only if b ∈ VMOA, and in the Schatten class if and only if b ∈e B_p, 0 < p < ∞. This last result extends the analog theorem in the case of the unit disc of Peller [19] and Semmes [21]. In order to characterize the bounded Hankel operators, we prove a factorization theorem for functions in H¹ (Ω), a result that is of independent interest.     

Linear combination of composition operators on $$H^\infty $$ H ∞ and the Bloch space

Archiv der Mathematik - Let $$\lambda _i (i=1,\ldots ,k)$$ be nonzero complex scalars and $$\varphi _i (i=1,..,k)$$ be analytic self-maps of the unit disk $$\mathbb {D}$$ . We show that the...     

Boundedness from below of composition operators on the Bloch spaces

This paper proves that a necessary condition for a composition operator on the Bloch space of the unit disk to be bounded below given by Ghatage, Yan and Zheng is also sufficient. Given furthermore are a sufficient condition and a necessary condition of the boundedness from below for a composition operator on the Bloch space of a unit ball with dimensions bigger than one.     

Spectrum of Normal Weighted Composition Operators on the Fock Space Over ℂN

By using commutative C^*-algebra techniques, spectrum and essential spectrum of normal weighted composition operators on the Fock space over C^N are completely characterized. As an application, spectrum of self-adjoint weighted composition operators on the Fock space are obtained also.     

On the injectivity of twisted spherical means on ℂ n

In this paper we study the injectivity and injectivity sets for the twisted spherical means on ℂ ⁿ . These results are then used to study the same problems on the reduced Hiesenberg group.     

Nonclassical weighted estimates of certain Calderónzygmund operators on the plane

Let F be a compact subset of ? let μ be a Borel measure on F, and let ρ(z) be the distance of z to F. Denote $$A_K (f)(z) = \int\limits_F {K(\varsigma ,z)f(\varsigma ) dm(\varsigma ), z \in \mathbb{C}\backslash F}$$ where K (ζ, z) is either (ζ-z)² or (|ζ-z|(ζ-z))^-1 and m is the Lebesgue measure. Let ψ be a monotone nondecreasing positive function on (0, ∞) and let Φ(z)=Ψ(ρ(z))ρ(z), z ε ?/F. Under some additional assumptions on μ, it is proved that A_K is bounded from L² (μ) to L² (Φm) if and only if $$\int\limits_0^{ + 1} {\tfrac{{\psi (t)}}{t} + \int\limits_1^\infty {\tfrac{{\psi (t)}}{{t^2 }}dt< \infty } }$$ Thus, no interference of values of K of various signs is observed in such a situation. Bibliography: 4 titles.     

Boundedness of composition of operators associated with different homogeneities on weighted Besov and Triebel–Lizorkin spaces

In this paper, we introduce weighted Besov spaces and weighted Triebel–Lizorkin spaces associated with different homogeneities and prove that the composition of two Calderón–Zygmund operators is bounded on these spaces. This extends a recent result in Han et al, Revista Mat. Iber.