Boundary behavior in Hilbert spaces of vector-valued analytic functions

In this paper we study the boundary behavior of functions in Hilbert spaces of vector-valued analytic functions on the unit disc D. More specifically, we give operator-theoretic conditions on Mz, where Mz denotes the operator of multiplication by the identity function on D, that imply that all functions in the space have non-tangential limits a.e., at least on some subset of the boundary. The main part of the article concerns the extension of a theorem by Aleman, Richter and Sundberg in [A. Aleman, S. Richter, C. Sundberg, Analytic contractions and non-tangential limits, Trans. Amer. Math. Soc. 359 (2007)] to the case of vector-valued functions.     

Notes on ordered reproducing Hilbert spaces over the complex plane

Let f, g be entire functions. If there exist M₁,M₂>0 such that |f(z)|?M₁|g(z)| whenever |z|>M₂ we say that f?g. Let X be a reproducing Hilbert space with an orthogonal basis . We say that X is an ordered reproducing Hilbert space (or X is ordered) if f?g and g∈X imply f∈X. In this note, we show that if then X is ordered; if then X is not ordered. In the case , there are examples to show that X can be of order or opposite.     

Normal families and shared values of meromorphic functions

Let be a positive integer, let F be a family of meromorphic functions in a domain D, all of whose zeros have multiplicity at least k+1, and let , be two holomorphic functions on D. If, for each f∈F, f=a(z)⇔f(k)=h(z), then F is normal in D.     

Remarks on common hypercyclic vectors

We treat the question of existence of common hypercyclic vectors for families of continuous linear operators. It is shown that for any continuous linear operator T on a complex Fréchet space X and a set Λ⊆R₊×C which is not of zero three-dimensional Lebesgue measure, the family has no common hypercyclic vectors. This allows to answer negatively questions raised by Godefroy and Shapiro and by Aron. We also prove a sufficient condition for a family of scalar multiples of a given operator on a complex Fréchet space to have a common hypercyclic vector. It allows to show that if and φ∈H^∞(D) is non-constant, then the family has a common hypercyclic vector, where Mφ:H²(D)→H²(D), Mφf=φf, and , providing an affirmative answer to a question by Bayart and Grivaux. Finally, extending a result of Costakis and Sambarino, we prove that the family has a common hypercyclic vector, where Tbf(z)=f(z−b) acts on the Fréchet space H(C) of entire functions on one complex variable.     

Abstract splines in Krein spaces

We present generalizations to Krein spaces of the abstract interpolation and smoothing problems proposed by Atteia in Hilbert spaces: given a Krein space K and Hilbert spaces H and E (bounded) surjective operators T:H→K and V:H→E, ρ>0 and a fixed z₀∈E, we study the existence of solutions of the problems and .     

On classes of analytic functions related to conic domains

We introduce classes of analytic functions related to conic domains, using a new linear multiplier fractional differential operator (n∈N₀={0,1,…}, 0?α<1, λ?0), which is defined as

D⁰f(z)=f(z),     

Stable functions and Vietoris' theorem

An analytic function f(z) in the unit disc D is called stable if s_n(f,·)/f?1/f holds for all for . Here s_n stands for the nth partial sum of the Taylor expansion about the origin of f, and ? denotes the subordination of analytic functions in . We prove that (1−z)^λ, λ∈[−1,1], are stable. The stability of turns out to be equivalent to a famous result of Vietoris on non-negative trigonometric sums. We discuss some generalizations of these results, and related conjectures, always with an eye on applications to positivity results for trigonometric and other polynomials.     

Univalence and starlikeness of certain transforms defined by convolution of analytic functions

Let U(λ) denote the class of all analytic functions f in the unit disk Δ of the form f(z)=z+a₂z²+? satisfying the condition

     

Homogeneous multiplication operators on bounded symmetric domains

Let D=G/K be an irreducible bounded symmetric domain of dimension d and let be the analytic continuation of the weighted Bergman spaces of holomorphic functions on D. We consider the d-tuple M=(M₁,…,M_d) of multiplication operators by coordinate functions and consider its spectral properties. We find those parameters ν for which the tuple M is subnormal and we answer some open questions of Bagchi and Misra. In particular, we prove that when D=B^d is the unit ball in , then B^d is a k-spectral set of M if and only if is the Hardy space or a weighted Bergman space.     

Normal families of meromorphic functions with multiple zeros and poles

Let k be a positive integer with k?2 and let be a family of functions meromorphic on a domain D in , all of whose poles have multiplicity at least 3, and of whose zeros all have multiplicity at least k+1. Let a(z) be a function holomorphic on D, a(z)?0. Suppose that for each , f^(k)(z)≠a(z) for z∈D. Then is a normal family on D.     

Composition operators from the Hardy space to the nth weighted-type space on the unit disk and the half-plane

The boundedness of the composition operator C_φf(z)=f(φ(z)) from the Hardy space , where X is the upper half-plane or the unit disk D={z∈C:|z|<1} in the complex plane C, to the nth weighted-type space, where φ is an analytic self-map of X, is characterized.     

A constructive and functorial embedding of locally compact metric spaces into locales

The paper establishes, within constructive mathematics, a full and faithful functor M from the category of locally compact metric spaces and continuous functions into the category of formal topologies (or equivalently locales). The functor preserves finite products, and moreover satisfies f?g if, and only if, M(f)?M(g) for continuous . This makes it possible to transfer results between Bishop's constructive theory of metric spaces and constructive locale theory.     

Proper analytic free maps

This paper concerns analytic free maps. These maps are free analogs of classical analytic functions in several complex variables, and are defined in terms of non-commuting variables amongst which there are no relations - they are free variables. Analytic free maps include vector-valued polynomials in free (non-commuting) variables and form a canonical class of mappings from one non-commutative domain D in say g variables to another non-commutative domain in variables. As a natural extension of the usual notion, an analytic free map is proper if it maps the boundary of D into the boundary of . Assuming that both domains contain 0, we show that if is a proper analytic free map, and f(0)=0, then f is one-to-one. Moreover, if also , then f is invertible and f−1 is also an analytic free map. These conclusions on the map f are the strongest possible without additional assumptions on the domains D and .     

Region of variability of two subclasses of univalent functions

Let F₁ (F₂ respectively) denote the class of analytic functions f in the unit disk |z|<1 with f(0)=0=f^′(0)−1 satisfying the condition RePf(z)<3/2 (RePf(z)>−1/2 respectively) in |z|<1, where Pf(z)=1+zf^″(z)/f^′(z). For any fixed z₀ in the unit disk and λ∈[0,1), we shall determine the region of variability for logf^′(z₀) when f ranges over the class and , respectively.     

Weighted composition operators from area Nevanlinna spaces into Bloch spaces

Let H(D) denote the class of all analytic functions on the open unit disk D of C. Let φ be an analytic self-map of D and u∈H(D). The weighted composition operator is defined by

     

About a condition for starlikeness

A condition for starlikeness will be improved given by the inequality Re(f^′(x)+αzf^″(z))>0, z∈U, concerning analytic functions of the form f(z)=z+a₂z²+? which are defined on the unit disk .     

BiBloch mappings and composition operators from Bloch type spaces to BMOA

The problem of constructing functions f₁, f₂ analytic in the unit disc D of the complex plane satisfying

     

COMPOSITION OPERATORS BETWEEN SOME WEIGHTED HARDY SPACES

Let D be the unit disc and H(D) be the set of all analytic functions on D. In [2], C. Cowen defined a space H = f ∈ H(D) : f(z) =sum from k=o to ∞ ak(z + 1)k, z∈ D, ‖f‖2 = sum from k=o to ∞ |ak|24k < ∞In this article, the authors consider the similar Hardy spaces with arbitrary weights and discuss some properties of them. Boundedness and compactness of composition operators between such spaces are also studied.