Hilbert Space Of Analytic Functions Research Articles

Composition-differentiation operators acting on certain Hilbert spaces of analytic functions

We study composition-differentiation operators acting on the Bergman and Dirichlet space of the unit disk. We first characterize the compactness of this operator on weighted Bergman spaces. We shall then prove that for an analytic self-map $\varphi$ on the unit disk $\disk$, the induced composition-differentiation operator is bounded with dense range if and only if $\varphi$ is univalent and the polynomials are dense in the Bergman space on $\varphi(\disk)$.

Cesàro-type operators associated with Borel measures on the unit disc acting on some Hilbert spaces of analytic functions

Given a complex Borel measure μ on the unit disc D={z∈C:|z|<1}, we consider the Cesàro-type operator Cμ defined on the space Hol(D) of all analytic functions in D as follows:If f∈Hol(D), f(z)=∑n=0∞anzn (z∈D), then Cμ(f)(z)=∑n=0∞μn(∑k=0nak)zn, (z∈D), where, for n≥0, μn denotes the n-th moment of the measure μ, that is, μn=∫Dwndμ(w).We study the action of the operators Cμ on some Hilbert spaces of analytic function in D, namely, the Hardy space H2 and the weighted Bergman spaces Aα2 (α>−1). Among other results, we prove that, if we set Fμ(z)=∑n=0∞μnzn (z∈D), then Cμ is bounded on H2 or on Aα2 if and only if Fμ belongs to the mean Lipschitz space Λ1/22. We prove also that Cμ is a Hilbert-Schmidt operator on H2 if and only if Fμ belongs to the Dirichlet space D, and that Cμ is a Hilbert-Schmidt operator on Aα2 if and only if Fμ belongs to the Dirichlet-type space D−1−α2.

Complex symmetric generalized weighted composition operators on Hilbert spaces of analytic functions

We investigate complex symmetry of a generalized weighted composition operator Dn,v,ψ on the reproducing kernel Hilbert space Hγ of analytic functions on the unit disk D. We first characterize a class of anti-linear weighted composition operators that are conjugations with a new approach. Then we obtain necessary and sufficient conditions for Dn,v,ψ to be complex symmetric with respect to these conjugations. Our results not only generalize and unify the ones in the literature, but also provide an affirmative answer to a question from a paper of Lim and Khoi (2018) [16]. The classes of self-adjoint, normal, unitary generalized weighted composition operators on Hγ and their relations with the property of complex symmetry are also studied.

A new characterization of the Hardy space and of other Hilbert spaces of analytic functions

A new characterization of the Hardy space and of other Hilbert spaces of analytic functions

Left-Invertibility of Rank-One Perturbations

For each isometry V acting on some Hilbert space and a pair of vectors f and g in the same Hilbert space, we associate a nonnegative number c(V; f, g) defined by $$\begin{aligned} c(V; f,g) = (\Vert f\Vert ^2 - \Vert V^*f\Vert ^2) \Vert g\Vert ^2 + |1 + \langle V^*f , g\rangle |^2. \end{aligned}$$We prove that the rank-one perturbation \(V + f \otimes g\) is left-invertible if and only if $$\begin{aligned} c(V;f,g) \ne 0. \end{aligned}$$We also consider examples of rank-one perturbations of isometries that are shift on some Hilbert space of analytic functions. Here, shift refers to the operator of multiplication by the coordinate function z. Finally, we examine \(D + f \otimes g\), where D is a diagonal operator with nonzero diagonal entries and f and g are vectors with nonzero Fourier coefficients. We prove that \(D + f\otimes g\) is left-invertible if and only if \(D+f\otimes g\) is invertible.

Specht Oranına Göre Berezin Sayı Eşitsizlikleri

Smooth functions are associated with operators on Hilbert spaces of analytic functions through the Berezin transform. The Berezin symbol and the Berezin number of an operator A on the Hilbert functional space H(Ω) over some set Ω with the reproducing kernel are defined, respectively, by A ̃(μ)=〈A K_μ/K_μ ,K_μ/K_μ 〉,μ∈Ω and ber(A)=sup┬(μ∈Ω)⁡|A ̃(μ)|. By using this bounded function A ̃, we present some new Berezin number inequalities of Hilbert functional space operators. Some inequalities with respect to Specht's ratio are improved and generalized. Using these modifications, we also establish various new inequalities for the Berezin radius and Berezin norm of operators.

Random Weighted Shifts on Hilbert Spaces of Analytic Functions

Random Weighted Shifts on Hilbert Spaces of Analytic Functions

Multiplicative property of series used in the Nevanlinna-Pick problem

In the paper we obtained substantially new sufficient condition for negativity of coefficients of power series inverse to series with positive ones. It has been proved that element-wise product of power series retains this property. In particular, it gives rise to generalization of the classical Hardy theorem about power series. These results are generalized for cases of series with multiple variables. Such results are useful in Nevanlinna – Pick theory. For example, if function k(x, y) can be represented as power series Pn≥0 an(x¯y)n, an &gt; 0, and reciprocal function 1/k(x, y) can be represented as power series Pn≥0 bn(x¯y)n such that bn &lt; 0, n &gt; 0, then k(x, y) is a reproducing kernel function for some Hilbert space of analytic functions in the unit disc D with Nevanlinna – Pick property. The reproducing kernel 1/(1 − x¯y) of the classical Hardy space H2(D) is a prime example for our theorems.

Multiplier tests and subhomogeneity of multiplier algebras

Multipliers of reproducing kernel Hilbert spaces can be characterized in terms of positivity of $n \times n$ matrices analogous to the classical Pick matrix. We study for which reproducing kernel Hilbert spaces it suffices to consider matrices of bounded size $n$. We connect this problem to the notion of subhomogeneity of non-selfadjoint operator algebras. Our main results show that multiplier algebras of many Hilbert spaces of analytic functions, such as the Dirichlet space and the Drury-Arveson space, are not subhomogeneous, and hence one has to test Pick matrices of arbitrarily large matrix size $n$. To treat the Drury-Arveson space, we show that multiplier algebras of certain weighted Dirichlet spaces on the disc embed completely isometrically into the multiplier algebra of the Drury-Arveson space.

Best kernel approximation in Bergman spaces

Let H be a reproducing kernel Hilbert space of analytic functions on the unit disk D. The best kernel approximation problem for H is the following: given any positive integer n and any function f∈H find the best norm approximation of f by a linear combination of no more than n kernel functions K(z,zk), 1≤k≤n. The purpose of this paper is to prove the existence of best kernel approximation for weighted Bergman spaces with standard weights.

A survey of optimal polynomial approximants, applications to digital filter design, and related open problems

In the last few years, the notion of optimal polynomial approximant has appeared in the mathematics literature in connection with the Hilbert spaces of analytic functions of one or more variables. In the 1970s, researchers in engineering and applied mathematics introduced least-squares inverses in the context of digital filters in signal processing. It turns out that in the Hardy space \(H^{2},\) these objects are identical. This paper is a survey of known results about optimal polynomial approximants. In particular, we will examine their connections with orthogonal polynomials and reproducing kernels in weighted spaces and digital filter design. We will also describe what is known about the zeros of optimal polynomial approximants, their rates of decay, and convergence results. Throughout the paper, we state many open questions that may be of interest.

Marcinkiewicz\u2013Zygmund Inequalities for Polynomials in Bergman and Hardy Spaces

We study the relationship between sampling sequences in infinite dimensional Hilbert spaces of analytic functions and Marcinkiewicz–Zygmund inequalities in subspaces of polynomials. We focus on the study of the Hardy space and the Bergman space in one variable because they provide two settings with a strikingly different behavior.

Power series of several variables with condition of logarithmical convexity

We obtain a new version of Hardy theorem about power series of several variables reciprocal to the power series with positive coefficients. We prove that if the sequence {as} = as1,s2,...,sn, ||s|| ≥ K satisfies condition of logarithmically convexity and the first coefficient a0 is sufficiently large then reciprocal power series has only negative coefficients {bs} = bs1,s2,...,sn, except b0,0,...,0 for any K. The classical Hardy theorem corresponds to the case K = 0, n = 1. Such results are useful in Nevanlinna - Pick theory. For example, if function k(x, y) can be represented as power series Σn≥0 an(x-y)n, an &gt; 0, and reciprocal function 1/k(x,y) can be represented as power series Σn≥0 bn(x-y)n such that bn &lt; 0, n &gt; 0, then k(x, y) is a reproducing kernel function for some Hilbert space of analytic functions in the unit disc D with Nevanlinna-Pick property. The reproducing kernel 1/1-x-y of the classical Hardy space H2(D) is a prime example for our theorems.

Simply connected topological spaces of weighted composition operators

Abstract In this paper, we prove that the topological spaces of nonzero weighted composition operators acting on some Hilbert spaces of analytic functions on the unit open ball are simply connected.

Comparison of singular numbers of composition operators on different Hilbert spaces of analytic functions

We compare the rate of decay of singular numbers of a given composition operator acting on various Hilbert spaces of analytic functions on the unit disk D. We show that for the Hardy and Bergman spaces, our results are sharp. We also give lower and upper estimates of the singular numbers of the composition operator with symbol the “cusp map” and the lens maps, acting on weighted Dirichlet spaces.

Co-Isometric Weighted Composition Operators on Hilbert Spaces of Analytic Functions

We obtain a necessary and sufficient condition for a weighted composition operator to be co-isometric on a general weighted Hardy space of analytic functions in the unit disk whose reproducing kernel has the usual natural form. This turns out to be equivalent to the property of being unitary. The result reveals a dichotomy identifying a specific family of weighted Hardy spaces as the only ones that support non-trivial operators of this kind.

Multiplication Operators with Adjoint of Cowen-Douglas Class Operator

We will consider multiplication operators on Hilbert spaces of analyticfunctions on a domain $\omega \subset \cmark $. We determine the commutants of certainmultiplication operators with adjoints in a Cowen-Douglas class operator.

Zeros of optimal polynomial approximants: Jacobi matrices and Jentzsch-type theorems

We study the structure of the zeros of optimal polynomial approximants to reciprocals of functions in Hilbert spaces of analytic functions in the unit disk. In many instances, we find the minimum possible modulus of occurring zeros via a nonlinear extremal problem associated with norms of Jacobi matrices. We examine global properties of these zeros and prove Jentzsch-type theorems describing where they accumulate. As a consequence, we obtain detailed information regarding zeros of reproducing kernels in weighted spaces of analytic functions.

Hilbert spaces of analytic functions with a contractive backward shift

We consider Hilbert spaces of analytic functions in the disk with a normalized reproducing kernel and such that the backward shift f(z)↦f(z)−f(0)z is a contraction on the space. We present a model for this operator and use it to prove the surprising result that functions which extend continuously to the closure of the disk are dense in the space. This has several applications, for example we can answer a question regarding reverse Carleson embeddings for these spaces. We also identify a large class of spaces which are similar to the de Branges–Rovnyak spaces and prove some results which are new even in the classical case.

Closed Range Weighted Composition Operators on $$H^{2}$$ H 2 and $$A_{\alpha }^{2}$$ A α 2

In this paper, first we investigate bounded below weighted composition operators $$C_{\psi ,\varphi }$$ on a Hilbert space of analytic functions. Then for $$\psi \in H^{\infty }$$ and a univalent map $$\varphi $$ , we characterize all closed range weighted composition operators $$C_{\psi ,\varphi }$$ on $$H^{2}$$ and $$A_{\alpha }^{2}$$ . Also we show that for $$\psi \in H^{\infty }$$ which is bounded away from zero near the unit circle, the weighted composition operator $$C_{\psi ,\varphi }$$ is bounded below on $$H^{2}$$ or $$A_{\alpha }^{2}$$ if and only if $$C_{\varphi }$$ has closed range. Moreover, we investigate invertible operators $$C_{\psi _{1},\varphi _{1}}C_{\psi _{2},\varphi _{2}}^{*}$$ and $$C_{\psi _{1},\varphi _{1}}^{*}C_{\psi _{2},\varphi _{2}}$$ on $$H^{2}$$ and $$A_{\alpha }^{2}$$ .