José Bonet

Universitat Politècnica de València | UPV ·  Institute for Pure and Applied Mathematics (IUMPA)

We survey some recent developments in the theory of Frechet spaces and of their duals. Among other things, Section 4 contains new, direct proofs of properties of, and results on, Fr ´ echet spaces with the density condition, and Section 5 gives an account of the modern theory of general K ¨ othe echelon and co-echelon spaces. The final section is d...

There are recent results concerning the boundedness and also unboundedness of Bergman projections on weighted spaces of the unit disc in special cases of rapidly decreasing weights, i.e. “large” Bergman spaces. The aim of our paper is to show that the cases of boundedness are largely exceptional: in general the Bergman projections are unbounded. In...

Some criteria for the continuity of Hausdorff operators on weighted Banach spaces of analytic functions with sup-norms are presented. The operator is defined in a different way on spaces of entire functions and on spaces of analytic functions on the disc. Both cases are analyzed. Our results complement recent work by Stylogiannis and Galanopoulos,...

In this survey we report about recent work on weighted Banach spaces of analytic functions on the unit disc and on the whole complex plane defined with sup-norms and operators between them. Results about the solid hull and core of these spaces and distance formulas are reviewed. Differentiation and integration operators, Cesaro and Volterra operato...

We characterize the weight sequences $(M_p)_p$ such that the class of ultra-differentiable functions ${\mathcal E}_{(M_p)}$ defined by imposing conditions on the derivatives of the function in terms of this sequence coincides with a class of ultradifferentiable functions ${\mathcal E}_{(\omega)}$ defined by imposing conditions on the Fourier Laplac...

The generalized Cesàro operators $$C_t$$ C t , for $$t\in [0,1]$$ t ∈ [ 0 , 1 ] , were first investigated in the 1980s. They act continuously in many classical Banach sequence spaces contained in $${{\mathbb {C}}}^{{{\mathbb {N}}}_0}$$ C N 0 , such as $$\ell ^p$$ ℓ p , $$c_0$$ c 0 , c , $$bv_0$$ b v 0 , bv and, as recently shown in Curbera et al. (...

The generalized Ces\`aro operators $C_t$, for $t\in [0,1]$, were first investigated in the 1980's. They act continuously in many classical Banach sequence spaces contained in $\mathbb{C}^{\mathbb{N}_0}$, such as $\ell^p$, $c_0$, $c$, $bv_0$, $bv$ and, as recently shown, \cite{CR4}, also in the discrete Ces\`aro spaces $ces(p)$ and their (isomorphic...

Several properties of weighted composition operators acting between weighted spaces of analytic functions with values on a Banach space are characterized. These results are applied to study weighted composition operators between weighted inductive and projective limits of spaces of vector-valued analytic functions. Operators acting on vector-valued...

We consider weighted Bergman spaces $$A_\mu ^1$$ A μ 1 on the unit disc as well as the corresponding spaces of entire functions, defined using non-atomic Borel measures with radial symmetry. By extending the techniques from the case of reflexive Bergman spaces, we characterize the solid core of $$A_\mu ^1$$ A μ 1 . Also, as a consequence of a chara...

We consider weighted Bergman spaces $A_\mu^1$ on the unit disc as well as the corresponding spaces of entire functions, defined using non-atomic Borel measures with radial symmetry. By extending the techniques from the case of reflexive Bergman spaces we characterize the solid core of $A_\mu^1$. Also, as a consequence of a characterization of solid...

People unequivocally employ reviews to decide on purchasing an item or an experience on the internet. In that regard, the growing significance and number of opinions have led to the development of methods to assess their sentiment content automatically. However, it is not straightforward for the models to create a consensus value that embodies the...

Various dynamical properties of the differentiation and Volterra-type integral operators on generalized Fock spaces are studied. We show that the differentiation operator is always supercyclic on these spaces. We further characterize when it is hypercyclic, power bounded and uniformly mean ergodic. We prove that the operator satisfies the Ritt's re...

We investigate dynamical properties such as topological transitivity, (sequential) hypercyclicity, and chaos for backward shift operators associated to a Schauder basis on LF-spaces. As an application, we characterize these dynamical properties for weighted generalized backward shifts on Köthe coechelon sequence spaces $k_p((v^{(m)})_{m\in\mathbb{N...

The Fr\'{e}chet (resp.\ (LB)) sequence spaces $ces(p+) := \cap_{r > p} ces(r), 1 \leq p < \infty $ (resp.\ $ ces (p-) := \cup_{ 1 < r < p} ces (r), 1 < p \leq \infty),$ are known to be very different to the classical sequence spaces $ \ell_ {p+} $ (resp., $ \ell_{p_{-}}).$ Both of these classes of non-normable spaces $ ces (p+), ces (p-)$ are defin...

Continuity, compactness, the spectrum and ergodic properties of the differentiation operator are investigated, when it acts in the Fréchet space of all Dirichlet series that are uniformly convergent in all half‐planes {s∈C|Res>ε} for each ε>0. The properties of the formal inverse of the differentiation are also investigated.

We prove sufficient conditions for the boundedness and compactness of Toeplitz operators Ta in weighted sup-normed Banach spaces Hv∞ of holomorphic functions defined on the open unit disc D of the complex plane; both the weights v and symbols a are assumed to be radial functions on D. In an earlier work by the authors it was shown that there exists...

We describe the proper closed invariant subspaces of the integration operator when it acts continuously on countable intersections and countable unions of weighted Banach spaces of holomorphic functions on the unit disc or the complex plane. Applications are given to the case of Korenblum type spaces and Hörmander algebras of entire functions.

We consider the Cauchy problem for a general class of parabolic partial differential equations in the Euclidean space ℝ N . We show that given a weighted L p -space $L_w^p({\mathbb {R}}^N)$ with 1 ⩽ p < ∞ and a fast growing weight w , there is a Schauder basis $(e_n)_{n=1}^\infty$ in $L_w^p({\mathbb {R}}^N)$ with the following property: given an ar...

The discrete Cesàro operator C acts continuously in various classical Banach sequence spaces within \( {\mathbb {C}}^{{\mathbb {N}}}.\) For the coordinatewise order, many such sequence spaces X are also complex Banach lattices [eg. \(c_0, \ell ^p \) for \( 1 < p \le \infty , \) and \( {{\text {ces}}}(p)\) for \( p \in \{ 0 \} \cup ( 1, \infty )\)]....

We characterize nuclear weighted composition operators W ψ , φ f = ψ ( f ∘ φ ) W_{\psi ,\varphi }f=\psi (f\circ \varphi ) on weighted Banach spaces of analytic functions with sup-norms. Consequences about nuclear composition operators on Bloch type spaces are presented. They extend previous work by Fares and Lefèvre on nuclear composition operators...

We characterize the boundedness and compactness of Toeplitz operators Ta with radial symbols a in weighted H∞-spaces Hv∞ on the open unit disc of the complex plane. The weights v are also assumed radial and to satisfy the condition (B) introduced by the second named author. The main technique uses Taylor coefficient multipliers, and the results are...

We prove sufficient conditions for the boundedness and compactness of Toeplitz operators $T_a$ in weighted sup-normed Banach spaces $H_v^\infty$ of holomorphic functions defined on the open unit disc $\mathbb{D}$ of the complex plane; both the weights $v$ and symbols $a$ are assumed to be radial functions on $\mathbb{D}$. In an earlier work by the...

The purpose of this seminar, which was presented at the Universitat Polit\`ecnica de Val\`encia in late 2012, is to explain several results concerning the bounded approximation property for Fr\'echet spaces. We give a full detailed proof of an important result due to Pe\l czy\'nski that asserts that every separable Fr\'echet space with the bounded...

A characterization of those points of the unit circle which belong to the spectrum of a composition operator \(C_{\varphi }\), defined by a rotation \(\varphi (z)=rz\) with \(|r|=1\), on the space \(H_0(\mathbb {D})\) of all analytic functions which vanish at 0, is given. Examples show that the spectrum of \(C_{\varphi }\) need not be closed. In th...

We describe the proper closed invariant subspaces of the integration operator when it acts continuously on countable intersections and countable unions of weighted Banach spaces of holomorphic functions on the unit disc or the complex plane. Applications are given to the case of Korenblum type spaces and H\"ormander algebras of entire functions.

Every separable complex Fréchet space with a continuous norm is isomorphic to a space of holomorphic functions - José Bonet

Continuity, compactness, the spectrum and ergodic properties of the differentiation operator are investigated, when it acts in the Fr\'echet space of all Dirichlet series that are uniformly convergent in all half-planes $\{s \in \mathbb{C} \ | \ {\rm Re} s > \varepsilon \}$ for each $\varepsilon>0$. The properties of the formal inverse of the diffe...

The discrete Cesàro (Banach) sequence spaces \( {{\text {ces}}}(r), 1< r < \infty ,\) have been thoroughly investigated for over 45 years. Not so for their dual spaces \( d (s) \cong ( {{\text {ces}}}(r))', \) with \( \frac{1}{s} + \frac{1}{r} = 1 ,\) which are somewhat unwieldy. Our aim is to undertake a further study of the spaces d(s) and of var...

Extending a result of Mashreghi and Ransford, we prove that every complex separable infinite dimensional Fr\'echet space with a continuous norm is isomorphic to a space continuously included in a space of holomorphic functions on the unit disc or the complex plane, which contains the polynomials as a dense subspace. As a consequence examples of nuc...

Every bounded composition operator \(C_{\psi }\) defined by an analytic symbol \(\psi \) on the complex plane when acting on generalized Fock spaces \(\mathcal {F}_{\varphi }^{p}, 1 \le p \le \infty \), and \(p=0\), is power bounded. Mean ergodic and uniformly mean ergodic bounded composition operators on these spaces are characterized in terms of...

We investigate dynamical properties such as topological transitivity, (sequential) hypercyclicity, and chaos for backward shift operators associated to a Schauder basis on LF-spaces. As an application, we characterize these dynamical properties for weighted generalized backward shifts on K\"othe coechelon sequence spaces $k_p((v^{(m)})_{m\in\mathbb...

Various dynamical properties of the differentiation and Volterra-type integral operators on generalized Fock spaces are studied. We show that the differentiation operator is always supercyclic on these spaces. We further characterize when it is hypercyclic, power bounded and uniformly mean ergodic. We prove that the operator satisfies the Ritt’s re...

The continuity, compactness and the spectrum of the Volterra integral operator Vg\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_g$$\end{document} with symbol an anal...

We consider the classical Cauchy problem for the linear heat equation and integrable initial data in the Euclidean space $\mathbb{R}^N$. In the case $N=1$ we show that given a weighted $L^p$-space $L_w^p(\mathbb{R})$ with $1 \leq p < \infty$ and a fast growing weight $w$, there is a Schauder basis $(e_n)_{n=1}^\infty$ in $L_ w^p(\mathbb{R})$ with t...

A characterization of those points in the unit disc which belong to the spectrum of a composition operator $C_{\varphi}$, defined by a rotation $\varphi(z)=rz$ with $|r|=1$, on the space $H_0(\mathbb{D})$ of all analytic functions on the unit disc which vanish at $0$, is given. Examples show that the point $1$ may or may not belong to the spectrum...

We determine various properties of the regular (LB)-spaces , , generated by the family of Banach sequence spaces . For instance, is a (DFS)-space which coincides with a countable inductive limit of weighted -spaces; it is also Montel but not nuclear. Moreover, and are isomorphic as locally convex Hausdorff spaces for all choices of . In addition, w...

The discrete Ces\`aro operator $ C $ acts continuously in various classical Banach sequence spaces within $ \mathbb{C}^{\mathbb{N}}.$ For the coordinatewise order, many such sequence spaces $ X $ are also complex Banach lattices (eg. $c_0, \ell^p $ for $ 1 < p \leq \infty , $ and $ ces (p)$ for $ p \in \{ 0 \} \cup ( 1, \infty )).$ In such Banach l...

In analogy to the notion of associated weights for weighted spaces of analytic functions with sup-norms, p-associated weights are introduced for spaces of entire p-integrable functions, 1 ≤ p < ∞. As an application, necessary conditions for the boundedness of composition operators acting between general Fock type spaces are proved.

Let v be a radial weight function on the unit disc or on the complex plane. It is shown that for each analytic function \(f_0\) in the Banach space \(H_v^\infty \) of all analytic functions f such that v|f| is bounded, the distance of \(f_0\) to the subspace \(H_v^0\) of \(H_v^\infty \) of all the functions g such that v|g| vanishes at infinity is...

We determine the solid hull for 2 < p < ∞ and the solid core for 1 < p < 2 of weighted Bergman spaces A μp , 1 < p < ∞, of analytic functions on the disk and on the whole complex plane, for a very general class of nonatomic positive bounded Borel measures μ. New examples are presented. Moreover, we show that the space A μp , 1 < p < 1, is solid if...

The Fréchet sequence spaces \( ces(p+) \) are very different to the Fréchet sequence spaces \( \ell _{p+}, 1 \le p < \infty ,\) that generate them, (Albanese et al. in J Math Anal Appl 458:1314–1323, 2018). The aim of this paper is to investigate various properties (eg. continuity, compactness, mean ergodicity) of certain linear operators acting in...

The Banach sequence spaces ces(p) are generated in a specified way via the classical spaces \( \ell _p, 1< p < \infty .\) For each pair \( 1< p,q < \infty \) the (p, q)-multiplier operators from ces(p) into ces(q) are known. We determine precisely which of these multipliers is a compact operator. Moreover, for the case of \( p = q \) a complete des...

Multiplication operators on weighted Banach spaces and locally convex spaces of continuous functions have been thoroughly studied. In this note, we characterize when continuous multiplication operators on a weighted Banach space and on a weighted inductive limit of Banach spaces of continuous functions are power bounded, mean ergodic or uniformly m...

The algebra of all Dirichlet series that are uniformly convergent in the half-plane of complex numbers with positive real part is investigated. When it is endowed with its natural locally convex topology, it is a non-nuclear Fréchet Schwartz space with basis. Moreover, it is a locally multiplicative algebra but not a Q -algebra. Composition operato...

A detailed investigation is made of the continuity, spectrum and mean ergodic properties of the Cesàro operator C when acting on the strong duals of power series spaces of innite type. There is a dramatic dierence in the nature of the spectrum of C depending on whether or not the strong dual space (which is always Schwartz) is nuclear.

It is shown that the monomials $\Lambda=(z^n)_{n=0}^{\infty}$ are a Schauder basis of the Fr\'echet spaces $A_+^{-\gamma}, \ \gamma \geq 0,$ that consists of all the analytic functions $f$ on the unit disc such that $(1-|z|)^{\mu}|f(z)|$ is bounded for all $\mu > \gamma$. Lusky \cite{L} proved that $\Lambda$ is not a Schauder basis for the closure...

We determine the solid hull and solid core of weighted Banach spaces $H_v^\infty$ of analytic functions $f$ such that $v|f|$ is bounded, both in the case of the holomorphic functions on the disc and on the whole complex plane, for a very general class of radial weights $v$. Precise results are presented for concrete weights on the disc that could n...

The Banach spaces ces(p),1<p<∞, were intensively studied by G. Bennett and others. The largest solid Banach lattice in CN which contains ℓp and which the Cesàro operator C:CN(long rightwards arrow)CN maps into ℓp is ces(p). For each 1≤p<∞, the (positive) operator C also maps the Fréchet space ℓp+=(n-ary intersection)q>pℓq into itself. It is shown t...

Given a non-negative weight $v$, not necessarily bounded or strictly positive, defined on a domain $G$ in the complex plane, we consider the weighted space $H_v^\infty(G)$ of all holomorphic functions on $G$ such that the product $v|f|$ is bounded in $G$ and study the question of when is such a space complete. We obtain both some necessary and some...

The spectrum of the Cesàro operator \(\mathsf {C}\), which is always continuous (but never compact) when acting on the classical Korenblum space and other related weighted Fréchet or (LB) spaces of analytic functions on the open unit disc, is completely determined. It turns out that such spaces are always Schwartz but, with the exception of the Kor...

Unlike for $\ell_p$, $1<p\leq\infty$, the discrete Ces\`aro operator $C$ does not map $\ell_1$ into itself. We identify precisely those weights $w$ such that $C$ does map $\ell_1(w)$ continuously into itself. For these weights a complete description of the eigenvalues and the spectrum of $C$ are presented. It is also possible to identify all $w$ su...

The discrete Ces\`aro operator $\mathsf{C}$ is investigated in the class of power series spaces $\Lambda_0(\alpha)$ of finite type. Of main interest is its spectrum, which is distinctly different when the underlying Fr\'echet space $\Lambda_0(\alpha)$ is nuclear as for the case when it is not. Actually, the nuclearity of $\Lambda_0(\alpha)$ is char...

We study weighted $H^\infty$ spaces of analytic functions on the open unit disc in the case of non-doubling weights, which decrease rapidly with respect to the boundary distance. We characterize the solid hulls of such spaces and give quite explicit representations of them in the case of the most natural exponentially decreasing weights.

In this paper the spectrum of composition operators on the space of real analytic functions is investigated. In some cases it is completely determined while in some other cases it is only estimated.

The Cesaro operator $\mathsf{C}$, when acting in the classical growth Banach spaces $A^{-\gamma}$ and $A_0^{-\gamma}$, for $\gamma > 0 $, of analytic functions on $\mathbb{D}$, is investigated. Based on a detailed knowledge of their spectra (due to A. Aleman and A.-M. Persson) we are able to determine the norms of these operators precisely. It is t...

Given a continuous, radial, rapidly decreasing weight $v$ on the complex plane $\mathbf{C}$, we study the solid hull of its associated weighted space $H_v^\infty(\mathbf{C})$ of all the entire functions $f$ such that $v|f|$ is bounded. The solid hull is found for a large class of weights satisfying the condition (B) of Lusky. Precise formulations a...

The classical spaces ℓ p+, 1 ≤ p < ∞, and L p−, 1<p ≤ ∞, are countably normed, reflexive Fréchet spaces in which the Cesàro operator C acts continuously. A detailed investigation is made of various operator theoretic properties of C (e.g., spectrum, point spectrum, mean ergodicity) as well as certain aspects concerning the dynamics of C (e.g., hype...

The spectrum and point spectrum of the Cesàro averaging operator (Formula presented.) acting on the Fréchet space (Formula presented.) of all (Formula presented.) functions on the interval (Formula presented.) are determined. We employ an approach via (Formula presented.)-semigroup theory for linear operators. A spectral mapping theorem for the res...

A detailed investigation is made of the continuity, the compactness and the spectrum of the Cesàro operator (Formula presented.) acting on the weighted Banach sequence space (Formula presented.) for a bounded, strictly positive weight w. New features arise in the weighted setting (e.g. existence of eigenvalues, compactness, mean ergodicity) which a...

An investigation is made of the continuity, the compactness and the spectrum of the Cesàro operator $\mathsf{C}$ when acting on the weighted Banach sequence spaces $\ell _{p}(w)$, $1 , for a positive decreasing weight $w$, thereby extending known results for $\mathsf{C}$ when acting on the classical spaces $\ell _{p}$. New features arise in the wei...

We investigate the spectrum of the Volterra operator $V_g$ with symbol an entire function $g$, when it acts on weighted Banach spaces $H_v^{\infty}(\mathbb{C})$ of entire functions with sup-norms and when it acts on H\"ormander algebras $A_p$ or $A^0_p$.

The abscissas of convergence, uniform convergence and absolute convergence of vector valued Dirichlet series with respect to the original topology and with respect to the weak topology $\sigma(X,X')$ of a locally convex space $X$, in particular of a Banach space $X$, are compared. The relation of their coincidence with geometric or topological prop...

We study the integration operator, the differentiation operator and more general differential operators on radial Fréchet or (LB) Hörmander algebras of entire functions. We analyze when these operators are power bounded, hypercyclic and (uniformly) mean ergodic.

Various properties of the (continuous) Cesàro operator (Formula presented.), acting on Banach and Fréchet spaces of continuous functions and (Formula presented.)-spaces, are investigated. For instance, the spectrum and point spectrum of (Formula presented.) are completely determined and a study of certain dynamics of (Formula presented.) is underta...

We characterize boundedness, compactness and weak compactness of Volterra operators acting between different weighted Banach spaces of entire functions with weighted sup-norms in terms of the symbol g. Thus we complement recent work by Bassallote, Contreras, Hern\'andez-Mancera, Mart\'in and Paul for spaces of holomorphic functions on the disc and...

Frames and Bessel sequences in Fréchet spaces and their duals are defined and studied. Their relation with Schauder frames and representing systems is analyzed. The abstract results presented here, when applied to concrete spaces of analytic functions, give many examples and consequences about sampling sets and Dirichlet series expansions.

Let ε(ω) (IRN) denote the non-quasianalytic class of ω-ultradifferentiable functions of Beurling type on IRN. Using a characterization of the bornological linear subspaces of (DFS)-spaces, an idea of Carleson and Hörmander's solution of the -problem, it is characterized for which weight functions σ a certain Köthe sequence space Λ(σ, N) is containe...

Let (T(t)) t≥0 be a strongly continuous C 0 -semigroup of bounded linear operators on a Banach space X such that lim t→∞ ||T(t)/t||=0. Characterizations of when (T(t)) t≥0 is uniformly mean ergodic, i.e., of when its Cesàro means r -1 ∫ 0 r T(s)ds converge in operator norm as r→∞, are known. For instance, this is so if and only if the infinitesimal...

Characterizations of interpolating multiplicity varieties for Hörmander algebras ${A_p(\mathbb{C})}$ and ${A^0_p(\mathbb{C})}$ of entire functions were obtained by Berenstein and Li (J Geom Anal 5(1):1–48, 1995) and Berenstein et al. (Can J Math 47(1):28–43, 1995) for a radial subharmonic weight p with the doubling property. In this note we con...

We obtain full description of eigenvalues and eigenvectors of composition operators \({C_{\varphi}:\fancyscript{A}\mathbb{R}\to \fancyscript{A}\mathbb{R}}\) for a real analytic self map \({\varphi:\mathbb{R} \rightarrow \mathbb{R} }\) as well as an isomorphic description of corresponding eigenspaces. We completely characterize those \({\varphi}\) f...

Well-known Banach space results (e.g., due to J. Koliha and Y. Katznelson/L. Tzafriri), which relate conditions on the spectrum of a bounded operator T to the operator norm convergence of certain sequences of operators generated by T , are extended to the class of quojection Fréchet spaces. These results are then applied to establish various me...

We characterize Kothe echelon spaces (and, more generally, those Frechet spaces with an unconditional basis) which are Schwartz, in terms of the convergence of the Cesaro means of power bounded operators defined on them. This complements similar known characterizations of reflexive and of Frechet-Montel spaces with a basis. Every strongly convergen...

We study the operators of differentiation and of integration and the Hardy operator on weighted Banach spaces of entire functions. We estimate the norm of the operators, study the spectrum, and analyze when they are surjective, power bounded, hypercyclic, and (uniformly) mean ergodic.

Various properties of the (continuous) Ces\`aro operator $\mathsf{C}$, acting on Banach and Fr\'echet spaces of continuous functions and $L^p$-spaces, are investigated. For instance, the spectrum and point spectrum of $\mathsf{C}$ are completely determined and a study of certain dynamics of $\mathsf{C}$ is undertaken (eg. hyper- and supercyclicity,...

We investigate the surjectivity of the Borel map in the quasianalytic setting for classes of ultradifferentiable functions defined in terms of the growth of the Fourier-Laplace transform. We deal with both the Roumieu E {ω} and the Beurling E (ω) classes for a weight function ω. In particular, we show that a classical result of Carleman for the qua...

We characterize the entire functions which transform a weighted Banach space of holomorphic functions on the disc of type $H^{\infty }$ into another such space by superposition. We also show that all the superposition operators induced by such entire functions map bounded sets into bounded sets and are continuous. Superposition operators that map...

For C 0-semigroups of continuous linear operators acting in a Banach space criteria are available which are equivalent to uniform mean ergodicity of the semigroup, meaning the existence of the limit (in the operator norm) of the Cesàro or Abel averages of the semigroup. Best known, perhaps, are criteria due to Lin, in terms of the range of the infi...

Every Köthe echelon Fréchet space XX that is Montel and not isomorphic to a countable product of copies of the scalar field admits a power bounded continuous linear operator TT such that I−TI−T does not have closed range, but the sequence of arithmetic means of the iterates of TT converges to 0 uniformly on the bounded sets in XX. On the other hand...

Motivated by recent work on the rate of growth of frequently hypercyclic entire functions due to Blasco, Grosse-Erdmann and Bonilla, we investigate conditions to ensure that the differentiation operator is chaotic or frequently hypercyclic on generalized weighted Bergman spaces of entire functions studied by Lusky, whenever the differentiation oper...

We study atomic decompositions in Fr\'echet spaces and their duals, as well as perturbation results. We define shrinking and boundedly complete atomic decompositions on a locally convex space, study the duality of these two concepts and their relation with the reflexivity of the space. We characterize when an unconditional atomic decomposition is s...

We investigate weighted composition operators that attain their norm on weighted Banach spaces of holomorphic functions on the unit disc of type H ∞. Applications for composition operators on weighted Bloch spaces are given.

We present criteria for determining mean ergodicity of C 0-semigroups of linear operators in a sequentially complete, locally convex Hausdorff space X. A characterization of reflexivity of certain spaces X with a basis via mean ergodicity of equicontinuous C 0-semigroups acting in X is also presented. Special results become available in Grothendiec...

We investigate several properties of operator-weighted composition maps Wψ,φ:f → ψ (f o φ) on unweighted H(D, X) and weighted H∞ν (D, X) spaces of vector-valued analytic functions on the unit disc D. Here φ is an analytic self-map of D and ψ is an analytic operator-valued function on D. We characterize when the operator is continuous, maps a neighb...

We study the dynamical behaviour of composition operators C ϕ defined on spaces A (Ω) of real analytic functions on an open subset Ω of R d . We characterize when such operators are topologically transitive, i.e. when for every pair of non-empty open sets there is an orbit intersecting both of them. Moreover, under mild assumptions on the compositi...

We exhibit examples of Fréchet Montel spaces E which have a non-reflexive Fréchet quotient but such that every Banach quotient is finite-dimensional. The construction uses a method developed by Albanese and Moscatelli and requires new ingredients. Some of the main steps in the proof are presented in Section 2. They are of independent interest and s...

We give a characterization, in one variable case, of those C∞C∞ multipliers F such that the division problem is solvable in S′(R)S′(R). For these functions F∈OM(R)F∈OM(R) we even prove that the multiplication operator MF(G)=FGMF(G)=FG has a continuous linear right inverse on S′(R)S′(R), in contrast to what happens in the several variables case, as...

In this note we show that weakly compact operators from a Banach space X into a complete (LB)-space E need not factorize through a reflexive Banach space. If E is a Fréchet space, then weakly compact operators from a Banach space X into E factorize through a reflexive Banach space. The factorization of operators from a Fréchet or a complete (LB)-sp...

Connections between (positive) mean ergodic operators acting in Banach lattices and properties of the underlying lattice itself are well understood (see the works of Emel'yanov, Wolff and Zaharopol). For Fréchet lattices (or more general locally convex solid Riesz spaces) there is virtually no information available. For a Fréchet lattice E, it is s...

We establish a criterion to decide when a countable projective limit of countable inductive limits of normed spaces is bornological. We compare the conditions occurring within our criterion with well-known abstract conditions from the context of homological algebra and with conditions arising within the investigation of weighted PLB-spaces of conti...

We characterize those composition operators defined on spaces of holomorphic functions of several variables which are power bounded, i.e. the orbits of all the elements are bounded. This condition is equivalent to the composition operator being mean ergodic. We also describe the form of the symbol when the composition operator is mean ergodic. Key...

We study the dynamical behaviour of composition operators defined on spaces of real analytic functions. We characterize when such operators are power bounded, i.e. when the orbits of all the elements are bounded. In this case this condition is equivalent to the composition operator being mean ergodic. In particular, we show that the composition ope...

We study weakly compact operators from a C*-algebra with values in a complete locally convex space. They constitute a natural non-commutative generalization of finitely additive vector measures with values in a locally convex space. Several results of Brooks, Sato and Wright are extended to this more general setting. Building on an approach due to...

Connections between (positive) mean ergodic operators acting in Banach lattices and properties of the underlying lattice itself are well understood (see the works of Emel'yanov, Wolff and Zaharopol). For Fréchet lattices (or more general locally convex solid Riesz spaces) there is virtually no information available. For a Fréchet lattice E, it is s...

We survey different developments on Fréchet, (LB)-spaces and (LF)-spaces of Moscatelli type, their properties, their applications to find counterexamples to different open problems, relevant examples of smooth function spaces in this frame and related results.

Answering a question of W. Arendt and M. Kunze in the negative, we construct a Banach space X and a norm closed weak* dense subspace Y of the dual X′ of X such that X, endowed with the Mackey topology μ(X,Y ) of the dual pair X,Y , is not complete.