Let B_E be the open unit ball of a complex finite- or infinite-dimensional Hilbert space. If f belongs to the space ℬ(B_E) of Bloch functions on B_E , we prove that the dilation map given by x ↦ (1-‖ x‖ ^2) ℛf(x) for x ∈ B_E , where ℛf denotes the radial derivative of f, is Lipschitz continuous with respect to the pseudohyperbolic distance ρ _E in B_E , which extends to the finite- and infinite-dimensional setting the result given for the classical Bloch space ℬ . To provide this result, we will need to prove that ρ _E(zx,zy) ≤ |z| ρ _E(x,y) for x,y ∈ B_E under some conditions on z ∈ℂ . Lipschitz continuity of x ↦ (1-‖ x‖ ^2) ℛf(x) will yield some applications on interpolating sequences for ℬ(B_E) which also extends classical results from ℬ to ℬ(B_E) . Indeed, we show that it is necessary for a sequence in B_E to be separated to be interpolating for ℬ(B_E) and we also prove that any interpolating sequence for ℬ(B_E) can be slightly perturbed and it remains interpolating.

Let B_E be the open unit ball of a complex finite or infinite dimensional Hilbert space E and consider the space ℬ(B_E) of Bloch functions on B_E . Using Lipschitz continuity of the dilation map on B_E given by x ↦ (1-‖ x‖ ^2) ℛf(x) for x ∈ B_E , where ℛf denotes the radial derivative of f ∈ℬ(B_E) , we study when a composition operator on ℬ(B_E) is bounded below.

It is known that there exists a constant 0<Δ _1 < 1 such that any Δ _1 -separated sequence for the pseudohyperbolic distance in the open unit disk D of ℂ is interpolating for the classical Bloch space ℬ . We will prove that 0.8114< Δ _1 < 0.9785 and we will also generalize this result for Bloch type spaces ℬ_v_p for v_p(z)=(1-|z|^2)^p . In particular, we will provide a construction to calculate an estimate of the lower and upper bounds for the corresponding constant of separation Δ _p for these spaces. We also prove that Δ _p tends to 1 when p →∞ .

When we deal with H∞, it is known that c0−interpolating sequences are interpolating and it is sufficient to interpolate idempotents of ℓ∞ in order to interpolate the whole ℓ∞. We will extend these results to the frame of interpolating sequences for Bloch type spaces Bv∞ and study the connection between the interpolating operators on Bv∞ and Bv0. Furthermore, for some particular weights v, we will provide examples of interpolating sequences for Bv∞ whose constant of separation is as close to 0 as desired.

The space of Bloch functions on bounded symmetric domains is extended by considering Bloch functions f f on the unit ball B E B_E of finite and infinite dimensional complex Banach spaces in two different ways: by extending the classical Bloch space considering the boundness of ( 1 − ‖ x ‖ 2 ) ‖ f ′ ( x ) ‖ (1-\|x\|^2) \|f’(x)\| on B E B_E and by preserving the invariance of the correspondiing seminorm when we compose with automorphisms φ \varphi of B E B_E . We study the connection between these spaces proving that they are different in general and prove that all bounded analytic functions on B E B_{E} are Bloch functions in both ways.

Every element in the boundary of the group of invertibles of a Banach algebra is a topological zero divisor. We extend this result to the scope of topological rings. In particular, we define a new class of semi-normed rings, called almost absolutely semi-normed rings, which strictly includes the class of absolutely semi-valued rings, and prove that every element in the boundary of the group of invertibles of a complete almost absolutely semi-normed ring is a topological zero divisor. To achieve all these, we have to previously entail an exhaustive study of topological divisors of zero in topological rings. In addition, it is also well known that the group of invertibles is open and the inversion map is continuous and $$\mathbb {C}$$ -differentiable in a Banach algebra. We also extend these results to the setting of complete normed rings. Finally, this study allows us to generalize the point, continuous and residual spectra to the scope of Banach algebras.

When we deal with H, it is known that c0−interpolating sequences are interpolating and it is sufficient to interpolate idempotents of l∞ in order to interpolate the whole l∞. We will extend these results to the frame of interpolating sequences in the classical Bloch space B and we will provide new characterizations of interpolating sequences for B. For that, bearing in mind that B is isomorphic but not isometric to the bidual of the little Bloch space B0, we will prove that given an interpolating sequence (zn) for B, the second adjoint of the interpolating operator which maps f ∈ B0 to the sequence (f (zn))(1− |zn| )) can be perfectly identified with the corresponding interpolating operator from B onto l∞. Furthermore, we will prove that interpolating sequences for H are also interpolating for B. This yields us to provide examples of interpolating sequences which are ε−separated for the pseudohyperbolic distance for ε > 0 as small as we want such that ε cannot be increased.

We show that an interpolating sequence for the weighted Banach space of analytic functions on the unit ball of a Hilbert space is hyperbolically separated. In the case of the so-called standard weights, a sufficient condition for a sequence to be linear interpolating is given in terms of Carleson type measures. Other conditions to be linearly interpolating are provided as well. Our results apply to the space of Bloch functions of such unit ball.

Given a nonautonomous discrete dynamical system (NDS) (X, f1,∞) we show that transitivity and density of periodic points do not imply sensitivity in general, i.e., in the definition of Devaney chaos there are no redundant conditions for NDS.In addition, we show that if we also assume uniform convergence of the sequence (fn) that induces the NDS, then sensitivity follows.Furthermore, in contrast to the autonomous case, we show that there exist minimal NDS which are neither equicontinuous nor sensitive.

Every analytic self-map of the unit ball of a Hilbert space induces a bounded composition operator on the space of Bloch functions. Necessary and sufficient conditions for compactness of such composition operators are provided, as well as some examples that clarify the connections among such conditions.

We introduce the sequence (a_n) ⊂ (0,1] and prove that the asymptotic behaviour of ∑_k=1^n a_k is the same than π(n), the prime-counting function. We also obtain that π(n) ∼ n a_n and we estimate 1/a_n-n/π(n) showing that lim_n →∞1/a_n-n/π(n) is convergent.

We prove that under the extended Carleson's condition, a sequence (x_n) ⊂ B_H is linear interpolating for H^∞(B_H) for an infinite dimensional Hilbert space H. In particular, we construct the interpolating functions for each sequence and find a bound for the constant of interpolation.

We introduce the concept of *-mapping as a selection of the duality mapping. We prove that the *-mappings are more general than the support mappings and provide a simple proof of the characterisation of smoothness by the norm to weak-star continuity of the *-mappings. As a consequence, we provide a characterisation of Hilbert spaces in terms of *-mappings and show that a Banach space has the Schur property if and only if it has the Dunford-Pettis property and there exists a *-mapping that is sequentially w – w continuous at 0. This last fact leads to the existence of smooth Banach spaces on which the duality mapping is not sequentially w – w continuous at 0.

We characterize the points of ·-w^* continuity of dual maps, turning out to be the smooth points. We prove that a Banach space has the Schur property if and only if it has the Dunford-Pettis property and there exists a dual map that is sequentially w-w continuous at 0. As consequence, we show the existence of smooth Banach spaces on which the dual map is not w-w continuous at 0.

The Bloch space has been studied on the open unit disk of C and some homogeneous domains of Cn. We define Bloch functions on the open unit ball of a Hilbert space E and prove that the corresponding space B(BE) is invariant under composition with the automorphisms of the ball, leading to a norm that — modulo the constant functions — is automorphism invariant as well. All bounded analytic functions on BE are also Bloch functions.

We consider analytic self‐maps φ on \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbf {D}$\end{document} and prove that the composition operator Cφ acting on \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$H_{v}^0$\end{document} is hypercyclic if φ is an automorphism or a hyperbolic non‐automorphic symbol with no fixed point. We give examples of weights v and parabolic non‐automorphisms φ on \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbf {D}$\end{document} which yield non‐hypercyclic composition operators Cφ on \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$H_{v}^0$\end{document}.

We characterize compact sets of𝔼1endowed with the level convergence topologyτℓ. We also describe the completion(𝔼1̂,𝒰̂)of𝔼1with respect to its natural uniformity, that is, the pointwise uniformity𝒰, and show other topological properties of𝔼1̂, as separability. We apply these results to give an Arzela-Ascoli theorem for the space of(𝔼1,τℓ)-valued continuous functions on a locally compact topological space equipped with the compact-open topology.

The study of composition operators on various spaces of analytic functions has quite a long and rich history. Let us mention two reasons for this. First, composition operators appear naturally in a variety of problems, e.g. in the study of commutants of multiplication operators and in the theory of dynamical systems. Second, the theory of composition operators links quite basic questions you can ask about linear operators with classical results from complex analysis. Thus, the boundedness of composition operators Cφ acting on the Hardy space H is closely connected with Littlewood’s Subordination Principle while the study of compactness leads to the theorem of Julia-Carathéodory on angular derivatives. Composition operators have been studied by many authors on various spaces of analytic functions. See e.g. [7], [8], [9], [10], [11], [12] [13], [14], [22], [23], [24], [25] and the references therein. Since the literature on this topic is growing steadily this can only be a sample of articles. In this article we are interested in composition operators acting in the following setting. Let v : D → (0,∞) be a bounded and continuous function (weight). Then we consider

We extend some results related to composition operators on H v ( G ) to arbitrary linear operators on H v0 ( G ) and H v ( G ). We also give examples of rank-one operators on H v ( G ) which cannot be approximated by composition operators. Keywords: Weighted Banach spaces of holomorphic functions, Schur spaces, weakly compact operators, compact operators, property (V) Quaestiones Mathematicae 35(2012), 463–470

The theme of this paper is the study of the separability of subspaces of holomorphic functions respect to the convergence over a given set and its connection with the metrizability of the polynomial topology. A notion closely related to this matter is that of Asplund set. Our discussion includes an affirmative answer to a question of Globevnik about interpolating sequences. We also consider the interplay between polynomials and Asplund sets and derive some consequences of it. Among them we obtain a characterization of Radon-Nikodym composition operators on algebras of bounded analytic functions.