Spaces Of Holomorphic Functions Research Articles

Estimates on Some General Classes of Holomorphic Function Spaces

In this current manuscript, some general classes of weighted analytic function spaces in a unit disc are defined and studied. Special functions significant in both analytic T(p,q,m,s;Ψ) norms and analytic Ψ-Bloch norms serve as a framework for introducing new families of analytic classes. An application in operator theory is provided by establishing important properties of the composition-type operator Cϕ such as the boundedness and compactness with the help of the defined new classes.

Composition operators on holomorphic variable exponent spaces

We study composition operators in variable exponent Hölder space of holomorphic functions in the unit disc. We provide sufficient and, in some cases, necessary conditions for boundedness and for compactness of composition operators acting in these spaces.

Mean ergodic composition operators on spaces of holomorphic functions on a Banach space

We study mean ergodic composition operators on infinite dimensional spaces of holomorphic functions of different types when defined on the unit ball of a Banach or a Hilbert space: that of all holomorphic functions, that of holomorphic functions of bounded type and that of bounded holomorphic functions. Several examples in the different settings are given.

Hardy space of translated Dirichlet series

We study the Hardy space of translated Dirichlet series \({\mathscr {H}}_{+}\). It consists on those Dirichlet series \(\sum a_n n^{-s}\) such that for some (equivalently, every) \(1 \le p < \infty \), the translation \(\sum {a_{n}}n^{-(s+\frac{1}{\sigma })}\) belongs to the Hardy space \({\mathscr {H}}^{p}\) for every \(\sigma >0\). We prove that this set, endowed with the topology induced by the seminorms \(\left\{ \Vert \cdot \Vert _{2,k}\right\} _{k\in {\mathbb {N}}}\) (where \(\Vert \sum {a_{n}n^{-s}}\Vert _{2,k}\) is defined as \(\big \Vert \sum {a_n n^{-(s+\frac{1}{k})}} \big \Vert _{{\mathscr {H}}^{2}}\)), is a Fréchet space which is Schwartz and non nuclear. Moreover, the Dirichlet monomials \(\{n^{-s}\}_{n \in {\mathbb {N}}}\) are an unconditional Schauder basis of \({\mathscr {H}}_+\). We also explore the connection of this new space with spaces of holomorphic functions on infinite-dimensional spaces. In the spirit of Gordon and Hedenmalm’s work, we completely characterize the composition operator on the Hardy space of translated Dirichlet series. Moreover, we study the superposition operators on \({\mathscr {H}}_{+}\) and show that every polynomial defines an operator of this kind. We present certain sufficient conditions on the coefficients of an entire function to define a superposition operator. Relying on number theory techniques we exhibit some examples which do not provide superposition operators. We finally look at the action of the differentiation and integration operators on these spaces.

Multiplier and Composition Operator Between Several Holomorphic Function Spaces in $$\mathbf {C^{n}}$$

Multiplier and Composition Operator Between Several Holomorphic Function Spaces in $$\mathbf {C^{n}}$$

Weighted holomorphic mixed norm spaces in the unit disc defined in terms of Fourier coefficients

ABSTRACT Following the ideas of recent research in Karapetyants and Samko [Mixed norm variable exponent Bergman space on the unit disc. Complex Var Elliptic Equ. 2016;61:1090–1106], we continue the study of new weighted spaces of holomorphic functions on the unit disc with the mixed norm defined in terms of conditions on Fourier coefficients of a function. Here we present a general approach for the weighted case. We study the boundedness of the weighted Bergman projection and discuss boundedness and compactness of Toeplitz operators with radial symbols. We also provide a characterization of functions in these spaces.

Approximation by simple partial fractions in unbounded domains

For unbounded simply connected domains in the complex plane, bounded by several simple curves with regular asymptotic behaviour at infinity, we obtain necessary conditions and sufficient conditions for simple partial fractions (logarithmic derivatives of polynomials) with poles on the boundary of to be dense in the space of holomorphic functions in (with the topology of uniform convergence on compact subsets of ). In the case of a strip bounded by two parallel lines, we give estimates for the convergence rate to zero in the interior of of simple partial fractions with poles on the boundary of and with one fixed pole. Bibliography: 16 titles.

Concurrent universal Padé approximation

We prove concurrent universal Padé approximation for several universal Padé approximants of several types. Our results are generic in the space of holomorphic functions, in the space of formal power series as well as in a subspace of A∞. These results are valid for one center of expansion or for several centers as well. The Padé approximants allow generic approximation on arbitrary compact sets, not necessarily having connected component, in contrast with the partial sums of power series. We also establish affine genericity for a class of universal functions.

Constructive description of analytic Besov spaces in strictly pseudoconvex domains

We use the method of pseudoanalytic continuation to obtain a characterization of spaces of holomorphic functions with boundary values in Besov spaces in terms of polynomial approximations.

Path Components of the Space of (Weighted) Composition Operators on Bergman Spaces

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.

Characterization of hypercyclic weighted composition operators on the space of holomorphic functions

We characterize hypercyclic weighted composition operators acting on the Fréchet space of all holomorphic functions $H(\Omega )$. In particular we show that such operators exist when $\Omega $ is the punctured plane or the punctured disc.

On space of holomorphic functions with boundary smoothness and its dual

On space of holomorphic functions with boundary smoothness and its dual

Holomorphic function spaces on homogeneous Siegel domains

We study several connected problems of holomorphic function spaces on homogeneous Siegel domains. The main object of our study concerns weighted mixed norm Bergman spaces on homogeneous Siegel domains of type II. These problems include: sampling, atomic decomposition, duality, boundary values, boundedness of the Bergman projectors. Our analysis include the Hardy spaces, and suitable generalizations of the classical Bloch and Dirichlet spaces. One of the main novelties in this work is the generality of the domains under consideration, that is, homogeneous Siegel domains, extending many results from the more particular cases of the upper half-plane, Siegel domains of tube type over irreducible cones, or symmetric, irreducible Siegel domains of type II.

Disjointness of composition operators onHv0spaces

AbstractThe disjoint properties of finitely many composition operators acting on the weighted Banach spaces of holomorphic functions in the unit disk were investigated in this paper.

On the Dimension of the Bergman Space of Some Hartogs Domains with Higher Dimensional Bases

Let D be a Hartogs domain of the form $$D=D_{\varphi }(G)=\{(z,w)\in G\times \mathbb {C}^{N}\,:\,\Vert w\Vert <e^{-\varphi (z)}\}$$ , where $$\varphi $$ is a plurisubharmonic function on G and $$G\subseteq \mathbb {C}^{M}$$ is a pseudoconvex domain. We expand on the results of Jucha (J Geom Anal 22(1):23–37, 2012) and prove the infinite-dimensionality or triviality of the space of square integrable holomorphic functions on $$D_{\varphi }(G)$$ for various choices of $$\varphi $$ and G.

Reachable states and holomorphic function spaces for the 1-D heat equation

The description of the reachable states of the heat equation is one of the central questions in control theory. The aim of this work is to present new results for the 1-D heat equation with boundary control on the segment [0,π]. In this situation it is known that the reachable states are holomorphic in a square D the diagonal of which is given by [0,π]. The most precise results obtained recently say that the reachable space is contained between two well known spaces of analytic functions: the Smirnov space E2(D) and the Bergman space A2(D). We show that the space of reachable states is exactly the sum of two Bergman spaces on sectors the intersection of which is D. In order to get a more precise information on this sum of Bergman spaces, we also prove that it contains a certain weighted Bergman space on D.

On Some Classes of Weighted Spaces of Weakly Holomorphic Functions

Let v be a weight on a domain D in a metrizable locally convex space E and F be a complete locally convex space. Denote by Hv(D, F) the weighted space of F-valued holomorphic functions on D satisfying that v.f is bounded, Av(D) subspace of $H_{v}(D, \mathbb C)$ with the unit ball is compact for the open-compact topology. The aim of this paper is to study linearization theorems and approximation properties in several different topologies for weighted spaces Av(D, F) of functions f ∈ Hv(D, F) such that u ∘ f ∈ Av(D) for every continuous linear functional u on F.

Weighted/unweighted composition operators which are Ritt or unconditional Ritt operators

In this paper, we study when a composition operator or a weighted composition operator on a Banach space of holomorphic functions is a Ritt operator or an unconditional Ritt operator. It turns out that for composition operators or weighted composition operators on a Banach space of holomorphic functions, if a composition operator or a weighted composition operator is a Ritt operator, then it is also an unconditional Ritt operator.

Hadamard–Bergman Convolution Operators

We introduce a convolution form, in terms of integration over the unit disc $$\mathbb {D},$$ for operators on functions f in $$H(\mathbb {D})$$ , which correspond to Taylor expansion multipliers. We demonstrate advantages of the introduced integral representation in the study of mapping properties of such operators. In particular, we prove the Young theorem for Bergman spaces in terms of integrability of the kernel of the convolution. This enables us to refer to the introduced convolutions as Hadamard–Bergman convolution. Another, more important, advantage is the study of mapping properties of a class of such operators in Holder type spaces of holomorphic functions, which in fact is hardly possible when the operator is defined just in terms of multipliers. Moreover, we show that for a class of fractional integral operators such a mapping between Holder spaces is onto. We pay a special attention to explicit integral representation of fractional integration and differentiation.

Schiffer Comparison Operators and Approximations on Riemann Surfaces Bordered by Quasicircles

We consider a compact Riemann surface R of arbitrary genus, with a finite number of non-overlapping quasicircles, which separate R into two subsets: a connected Riemann surface Sigma , and the union mathcal {O} of a finite collection of simply connected regions. We prove that the Schiffer integral operator mapping the Bergman space of anti-holomorphic one-forms on mathcal {O} to the Bergman space of holomorphic forms on Sigma is an isomorphism onto the exact one-forms, when restricted to the orthogonal complement of the set of forms on all of R. We then apply this to prove versions of the Plemelj–Sokhotski isomorphism and jump decomposition for such a configuration. Finally we obtain some approximation theorems for the Bergman space of one-forms and Dirichlet space of holomorphic functions on Sigma by elements of Bergman space and Dirichlet space on fixed regions in R containing Sigma .