Spaces Of Holomorphic Functions Research Articles

Hypercyclic composition operators on the $$S^p $$ S p space with automorphism symbols

Let \(S^p\) be the space of holomorphic functions whose derivative lies in the classical Hardy space \(H^p\) over the unit disk. We prove in this paper that the composition operator \(C_\varphi \) with \(\varphi \) an automorphism is hypercyclic on \(S^p\), \(0<p<1\), if and only if \(\varphi \) has no interior fixed point. This answers affirmatively a problem posed by Colonna and Martinez-Avendano in the paper “Hypercyclicity of composition operators on Banach spaces of analytic functions” (Complex Anal Oper Theory 12(1): 305–323, 2018).

Interpolation property for hypercyclic elements of the Taylor Shift operator

In this article we deal with the hypercyclic complex Taylor shift operator acting on the space of holomorphic functions in a suitable simply connected domain, along with the real Taylor shift on the space of real infinitely differentiable functions. In both cases, we investigate a problem of interpolation for the class of hypercyclic elements. Further, we study the notion of disjoint hypercyclicity for subsequences of the orbit of the real Taylor shift.

Reproducing kernel functions and asymptotic expansions on Jordan-Kepler manifolds

We study the complex geometry of generalized Kepler manifolds, defined in Jordan theoretic terms, introduce Hilbert spaces of holomorphic functions defined by radial measures, and find the complete asymptotic expansion of the corresponding reproducing kernels for Kähler potentials, both in the flat and bounded setting.

Dynamics of weighted composition operators in the unit ball

In the present paper, we investigate the dynamic behavior of weighted composition operators acting on the space of holomorphic functions on the unit ball in CN.

Universality of Composition Operators and Applications to Holomorphic Dynamics

By investigating which level of universality composition operators Cf can have, where the symbol f is given by the restriction of a transcendental entire function to suitable parts of the Fatou set of f, this work combines the theory of dynamics of continuous linear operators on spaces of holomorphic functions with the theory of non-linear complex dynamics on the complex plane.

Generalized Hilbert Matrices Acting on Spaces that are Close to the Hardy Space $$H^1$$ H 1 and to the Space BMOA

It is known that if X and Y are spaces of holomorphic functions in the unit disc $$\mathbb {D}$$ , which are between the mean Lipschitz space $$\Lambda ^p_{1/p}$$ , where $$1<p<\infty $$ , and the Bloch space $$\mathcal {B}$$ , then the generalized Hilbert matrix $$\mathcal {H}_\mu $$ , induced by a positive Borel measure $$\mu $$ on the interval [0, 1), is a bounded operator from the space X into the space Y if and only if $$\mu $$ is a 1-logarithmic 1-Carleson measure. We improve this result by proving that the same conclusion holds if we replace the space $$\Lambda ^p_{1/p}$$ , $$1<p<\infty $$ , by the space $$\Lambda ^1_1$$ . Also we prove that the same conclusion holds if X and Y are spaces of holomorphic functions in $$\mathbb {D}$$ , which are between the Besov space $$\mathcal {B}^{1,1}$$ and the mixed norm space $$H^{\infty ,1,1}$$ . As immediate consequences, we obtain many results and some of them are new.

Paley–Wiener Theorems on the Siegel Upper Half-Space

In this paper we study spaces of holomorphic functions on the Siegel upper half-space $\mathcal U$ and prove Paley-Wiener type theorems for such spaces. The boundary of $\mathcal U$ can be identified with the Heisenberg group $\mathbb H_n$. Using the group Fourier transform on $\mathbb H_n$, Ogden-Vagi proved a Paley-Wiener theorem for the Hardy space $H^2(\mathcal U)$. We consider a scale of Hilbert spaces on $\mathcal U$ that includes the Hardy space, the weighted Bergman spaces, the weighted Dirichlet spaces, and in particular the Drury-Arveson space, and the Dirichlet space $\mathcal D$. For each of these spaces, we prove a Paley-Wiener theorem, some structure theorems, and provide some applications. In particular we prove that the norm of the Dirichlet space modulo constants $\dot{\mathcal D}$ is the unique Hilbert space norm that is invariant under the action of the group of automorphisms of $\mathcal U$.

Subspaces of Frequently Hypercyclic Functions for Sequences of Composition Operators

In this paper, a criterion for a sequence of composition operators defined on the space of holomorphic functions in a complex domain to be frequently hypercyclic is provided. Such criterion improves some already known special cases and, in addition, it is also valid to provide dense vector subspaces as well as large closed ones consisting entirely, except for zero, of functions that are frequently hypercyclic.

Surjectivity of isometries between weighted spaces of holomorphic functions

We examine the surjectivity of isometries between weighted spaces of holomorphic functions. We show that for certain classical weights on the open unit disc all isometries of the weighted space of holomorphic functions, $${ {\mathcal {H}}}_{v_o}( \varDelta )$$ , are surjective. Criteria for surjectivity of isometries of $${ \mathcal H}_v(U)$$ in terms of a separation condition on points in the image of $${ {\mathcal {H}}}_{v_o}(U)$$ are also given for U a bounded open set in $${\mathbb {C}}$$ . Considering the weight $$v(z)= 1-|z|^2$$ and the isomorphism $$f\mapsto f'$$ we are able to show that all isometries of the little Bloch space are surjective.

Small Bergman-Orlicz and Hardy-Orlicz spaces, and their composition operators

We show that the weighted Bergman-Orlicz space $A_{\alpha}^{\psi}$ coincides with some weighted Banach space of holomorphic functions if and only if the Orlicz function $\psi$ satisfies the so-called $\Delta^{2}$--condition. In addition we prove that this condition characterizes those $A_{\alpha}^{\psi}$ on which every composition operator is bounded or order bounded into the Orlicz space $L_{\alpha}^{\psi}$. This provides us with estimates of the norm and the essential norm of composition operators on such spaces. We also prove that when $\psi$ satisfies the $\Delta^{2}$--condition, a composition operator is compact on $A_{\alpha}^{\psi}$ if and only if it is order bounded into the so-called Morse-Transue space $M_{\alpha}^{\psi}$. Our results stand in the unit ball of $\mathbb{C}^{N}$.

Continuity and Holomorphicity of Symbols of Weighted Composition Operators

The main problem considered in this article is the following: if $\mathbf{F}$, $\mathbf{E}$ are normed spaces of continuous functions over topological spaces $X$ and $Y$ respectively, and $\omega:Y\to\mathbb{C}$ and $\Phi:Y\to X$ are such that the weighted composition operator $W_{\Phi,\omega}$ is continuous, when can we guarantee that both $\Phi$ and $\omega$ are continuous? An analogous problem is also considered in the context of spaces of holomorphic functions over (connected) complex manifolds. Additionally, we consider the most basic properties of the weighted composition operators, which only have been proven before for more concrete function spaces.

Dynamics of weighted composition operators on function spaces defined by local properties

We study topological transitivity/hypercyclicity and topological (weak) mixing for weighted composition operators on locally convex spaces of scalar-valued functions which are defined by local properties. As main application of our general approach we characterize these dynamical properties for weighted composition operators on spaces of ultradifferentiable functions, both of Beurling and Roumieu type, and on spaces of zero solutions of elliptic partial differential equations. Special attention is given to eigenspaces of the Laplace operator and the Cauchy-Riemann operator, respectively. Moreover, we show that our abstract approach unifies existing results which characterize hypercyclicity, resp. topological mixing, of (weighted) composition operators on the space of holomorphic functions on a simply connected domain in the complex plane, on the space of smooth functions on an open subset of $\mathbb{R}^d$, as well as results characterizing topological transitiviy of such operators on the space of real analytic functions on an open subset of $\mathbb{R}^d$.

Linear maps preserving inner functions

We show that, for many holomorphic function spaces on the unit disk, a continuous endomorphism that sends inner functions to inner functions is necessarily a weighted composition operator.

Interpolation and absolutely convergent series in Fréchet spaces

A theorem due to Eidelheit concerning the interpolation problem for a sequence of continuous linear functionals in a Fréchet space is generalized. A solvability criterion for the interpolation problem is obtained in the form of an absolutely convergent series whose elements are in a fixed set. A solution of the system of equations for a sequence of functionals is constructed explicitly in a particular case. These results are then applied to spaces of holomorphic functions. Bibliography: 15 titles.

Hyponormality of Toeplitz operators in several variables by the weighted shifts approach

ABSTRACT In this paper, we characterize hyponormality of a class of operators related to a tuple of weighted shifts. As applications we construct examples of hyponormal Toeplitz operators on Hilbert spaces of holomorphic functions on the polydisk or the unit ball which include the Bergman space of the polydisk, the Hardy space of the unit ball and the Drury-Arveson space.

Division subspaces and integrable kernels

In this note we prove that the reproducing kernel of a Hilbert space satisfying the division property has integrable form, is locally of trace class, and the Hilbert space itself is a Hilbert space of holomorphic functions.

Difference of Weighted Composition Operators on the Space of Cauchy Integral Transforms

In this paper, we provide a complete function theoretic characterizations for boundedness and compactness of difference of weighted composition operators from the space of Cauchy integral transforms to logarithmic weighted-type spaces. Surprisingly, an interesting feature of these characterizations is that they are free from pseudo-hyperbolic distance between $\varphi(z)$ and $\psi(z)$, which is different from the previous characterizations of difference of weighted composition operators acting between different holomorphic function spaces.

Generalized Hölder Spaces of Holomorphic Functions in Domains in the Complex Plane

We study some nonstandard spaces of functions holomorphic in domains on the complex plain with certain smoothness conditions up to the boundary. The first type is the space of Holder-type holomorphic functions with prescribed modulus of continuity $$\omega =\omega (h)$$ , and the second is the variable exponent holomorphic Holder space with the modulus of continuity $$|h|^{\lambda (z)}$$ . We give a characterization of functions in these spaces in terms of the behavior of their derivatives near the boundary.

English

In this article, we initiate the study of Bloch type spaces on the unit ball of a Hilbert space. As applications, the Hardy-Littlewood theorem in infinite-dimensional Hilbert spaces and characterizations of some holomorphic function spaces related to the Bloch type space are presented.

Difference of Differentiation Composition Operators on the Fractional Cauchy Transforms Spaces

In this paper, we consider the boundedness and compactness of the differences of differentiation composition operators from the space of fractional Cauchy transforms to the Bloch-type spaces and the weighted Dirichlet spaces. Surprisingly, these characterizations are free from pseudo-hyperbolic metric, which is a common feature of all the preceding characterizations of difference of differentiation composition operators on various spaces of holomorphic functions.