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Enrique Jordá

Publications and source records attributed to Enrique Jordá.

17 recordsLinked to original sources
arXiv2024

Positive definite functions as uniformly ergodic multipliers of the Fourier algebra

Let G be a locally compact group and let $ϕ$ be a positive definite function on G with $ϕ(e)=1$. This function defines a multiplication operator $M_ϕ$ on the Fourier algebra $A(G)$ of $G$. The aim of this paper is to classify the ergodic properties of the operators $M_ϕ$, focusing on several key factors, including the subgroup $H_ϕ=\{x\in G\colon ϕ(x)=1\}$, the spectrum of $M_ϕ$, or how ``spread-out'' a power of $M_ϕ$ can be. We show that the multiplication operator $M_ϕ$ is uniformly mean ergodic if and only if $H_ϕ$ is open and 1 is not an accumulation point of the spectrum of $M_ϕ$. Equivalently, this happens when some power of $ϕ$ is not far, in the multiplier norm, from a function supported on finitely many cosets of $H_ϕ$. Additionally, we show that the powers of $M_ϕ$ converge in norm if, and only if, the operator is uniformly mean ergodic and $H_ϕ=\{x\in G\colon |ϕ(x)|=1\}$.

math.FA↗
arXiv2024

Power boundedness and related properties for weighted composition operators on $\mathscr{S}(\mathbb{R}^d)$

We characterize those pairs $(ψ,φ)$ of smooth mappings $ψ:\mathbb{R}^d\rightarrow\mathbb{C},φ:\mathbb{R}^d\rightarrow\mathbb{R}^d$ for which the corresponding weighted composition operator $C_{ψ,φ}f=ψ\cdot(f\circφ)$ acts continuously on $\mathscr{S}(\mathbb{R}^d)$. Additionally, we give several easy-to-check necessary and sufficient conditions of this property for interesting special cases. Moreover, we characterize power boundedness and topologizablity of $C_{ψ,φ}$ on $\mathscr{S}(\mathbb{R}^d)$ in terms of $ψ,φ$. Among other things, as an application of our results we show that for a univariate polynomial $φ$ with $\text{deg}(φ)\geq 2$, power boundedness of $C_{ψ,φ}$ on $\mathscr{S}(\mathbb{R})$ for every $ψ\in\mathscr{O}_M(\mathbb{R})$ only depends on $φ$ and that in this case power boundedness of $C_{ψ,φ}$ is equivalent to $(C_{ψ,φ}^n)_{n\in\mathbb{N}}$ converging to $0$ in $\mathcal{L}_b(\mathscr{S}(\mathbb{R}))$ as well as to the uniform mean ergodicity of $C_{ψ,φ}$. Additionally, we give an example of a power bounded and uniformly mean ergodic weighted composition operator $C_{ψ,φ}$ on $\mathscr{S}(\mathbb{R})$ for which neither the multiplication operator $f\mapsto ψf$ nor the composition operator $f\mapsto f\circφ$ acts on $\mathscr{S}(\mathbb{R})$. Our results complement and considerably extend various results of Fernández, Galbis, and the second named author.

math.FA↗
arXiv2024

Cesàro operators associated with Borel measures acting on weighted spaces of holomorphic functions with sup-norm

Let $μ$ be a positive finite Borel measure on $[0,1).$ Cesàro-type operators $C_μ$ when acting on weighted spaces of holomorphic functions are investigated. In the case of bounded holomorphic functions on the unit disc we prove that $C_μ$ is continuous if and only if it is compact. In the case of weighted Banach spaces of holomorphic function defined by general weights, we give sufficient and necessary conditions for the continuity and compactness. For standard weights, we characterize the continuity and compactness on classical growth Banach spaces of holomorphic functions. We also study the point spectrum and the spectrum of $C_μ$ on the space of holomorphic functions on the disc, on the space of bounded holomorphic functions on the disc, and on the classical growth Banach spaces of holomorphic functions. All characterizations are given in terms of the sequence of moments $(μ_n)_{n\in\N_0}$. The continuity, compactness and spectrum of $C_μ$ acting on Fréchet and (LB) Korenblum type spaces are also considered.

math.FA↗
arXiv2024

Generalized Hilbert operators acting on weighted spaces of holomorphic functions with sup-norms

The behaviour of the generalized Hilbert operator associated with a positive finite Borel measure $μ$ on $[0,1)$ is investigated when it acts on weighted Banach spaces of holomorphic functions on the unit disc defined by sup-norms and on Korenblum type growth Banach spaces. It is studied when the operator is well defined, bounded and compact. To this aim, we study when it can be represented as an integral operator. We observe important differences with the behaviour of the Cesàro-type operator acting on these spaces, getting that boundedness and compactness are equivalent concepts for some standard weights. For the space of bounded holomorphic functions on the disc and for the Wiener algebra, we get also this equivalence, which is characterized in turn by the summability of the moments of the measure $μ.$ In the latter case, it is also equivalent to nuclearity. Nuclearity of the generalized Hilbert operator acting on related spaces, such as the classical Hardy space, is also analyzed.

math.FA↗
arXiv2023

Uniformly ergodic probability measures

Let $G$ be a locally compact group and $μ$ be a probability measure on $G$. We consider the convolution operator $λ_1(μ)\colon L_1(G)\to L_1(G)$ given by $λ_1(μ)f=μ\ast f$ and its restriction $λ_1^0(μ)$ to the augmentation ideal $L_1^0(G)$. Say that $μ$ is uniformly ergodic if the Cesàro means of the operator $λ_1^0(μ)$ converge uniformly to 0, that is, if $λ_1^0(μ)$ is a uniformly mean ergodic operator with limit 0 and that $μ$ is uniformly completely mixing if the powers of the operator $λ_1^0(μ)$ converge uniformly to 0. We completely characterize the uniform mean ergodicity of the operator $λ_1(μ)$ and the uniform convergence of its powers and see that there is no difference between $λ_1(μ)$ and $λ_1^0(μ) $ in this regard. We prove in particular that $μ$ is uniformly ergodic if and only if $G$ is compact, $μ$ is adapted (its support is not contained in a proper closed subgroup of $G$) and 1 is an isolated point of the spectrum of $μ$. The last of these three conditions is actually equivalent to $μ$ being spread-out (some convolution power of $μ$ is not singular). The measure $μ$ is uniformly completely mixing if and only if $G$ is compact, $μ$ is spread-out and the only unimodular value of the spectrum of $μ$ is 1.

math.FA↗
arXiv2020

Ergodic properties of convolution operators

Let $G$ be a locally compact group and $μ$ be a measure on $G$. In this paper we find conditions for the convolution operators $λ_p(μ)$, defined on $L^p(G)$ and given by convolution by $μ$, to be mean ergodic and uniformly mean ergodic. The ergodic properties of the operators $λ_p(μ)$ are related to the ergodic properties of the measure $μ$ as well.

math.FA↗
arXiv2019

Dynamics of weighted composition operators on spaces of continuous functions

Our study is focused on the dynamics of weighted composition operators defined on a locally convex space $E\hookrightarrow (C(X),τ_p)$ with $X$ being a topological Hausdorff space containing at least two different points and such that the evaluations $\{δ_x:\ x\in X\}$ are linearly independent in $E'$. We prove, when $X$ is compact and $E$ is a Banach space containing a nowhere vanishing function, that a weighted composition operator $C_{φ,ω}$ is never weakly supercyclic on $E$. We also prove that if the symbol $φ$ lies in the unit ball of $A(\mathbb{D})$, then every weighted composition operator can never be $τ_p$-supercyclic neither on $C(\mathbb{D})$ nor on the disc algebra $A(\mathbb{D})$. Finally, we obtain Ansari-Bourdon type results and conditions on the spectrum for arbitrary weakly supercyclic operators, and we provide necessary conditions for a composition operator to be weakly supercyclic on the space of holomorphic functions defined in non necessarily simply connected planar domains. As a consequence, we show that no composition operator can be weakly supercyclic neither on the space of holomorphic functions on the punctured disc nor in the punctured plane.

math.FA↗
arXiv2018

Spectrum of composition operators on ${\mathcal S}({\mathbb R})$ with polynomial symbols

We study the spectrum of operators in the Schwartz space of rapidly decreasing functions which associate each function with its composition with a polynomial. In the case where this operator is mean ergodic we prove that its spectrum reduces to 0, while the spectrum of any non mean ergodic composition operator with a polynomial always contains the closed unit disc except perhaps the origen. We obtain a complete description of the spectrum of the composition operator with a quadratic polynomial or a cubic polynomial with positive leading coefficient.

math.FA↗
arXiv2018

Tingley's problem for $p$-Schatten von Neumann classes

Let $H$ and $H'$ be a complex Hilbert spaces. For $p\in(1, \infty)\backslash\{2\}$ we consider the Banach space $C_p(H)$ of all $p$-Schatten von Neumann operators, whose unit sphere is denoted by $S(C_p(H))$. We prove that every surjective isometry $Δ: S(C_p(H))\to S(C_p(H'))$ can be extended to a complex linear or to a conjugate linear surjective isometry $T:C_p(H)\to C_p(H')$.

math.FA↗
arXiv2017

Dynamics and spectra of composition operators on the Schwartz space

In this paper we study the dynamics of the composition operators defined in the Schwartz space $\mathcal{S}(\mathbb{R})$ of rapidly decreasing functions. We prove that such an operator is never supercyclic and, for monotonic symbols, it is power bounded only in trivial cases. For a polynomial symbol $φ$ of degree greater than one we show that the operator is mean ergodic if and only if it is power bounded and this is the case when $φ$ has even degree and lacks fixed points. We also discuss the spectrum of composition operators.

math.FA↗
arXiv2016

Stability of derivations under weak-2-local continuous perturbations

Let $Ω$ be a compact Hausdorff space and let $A$ be a C$^*$-algebra. We prove that if every weak-2-local derivation on $A$ is a linear derivation and every derivation on $C(Ω,A)$ is inner, then every weak-2-local derivation $Δ:C(Ω,A)\to C(Ω,A)$ is a {\rm(}linear{\rm)} derivation. As a consequence we derive that, for every complex Hilbert space $H$, every weak-2-local derivation $Δ: C(Ω,B(H)) \to C(Ω,B(H))$ is a (linear) derivation. We actually show that the same conclusion remains true when $B(H)$ is replaced with an atomic von Neumann algebra. With a modified technique we prove that, if $B$ denotes a compact C$^*$-algebra (in particular, when $B=K(H)$), then every weak-2-local derivation on $C(Ω,B)$ is a (linear) derivation. Among the consequences, we show that for each von Neumann algebra $M$ and every compact Hausdorff space $Ω$, every 2-local derivation on $C(Ω,M)$ is a (linear) derivation.

math.OA↗
arXiv2016

Mean Ergodic Composition Operators on Banach spaces of holomorphic functions

Given a symbol $φ,$ i.e., a holomorphic endomorphism of the unit disc, we consider the composition operator $C_φ(f)=f\circφ$ defined on the Banach spaces of holomorphic functions $A(\mathbb{D})$ and $H^{\infty}(\mathbb{D})$. We obtain different conditions on the symbol $φ$ which characterize when the composition operator is mean ergodic and uniformly mean ergodic in the corresponding spaces. These conditions are related to the asymptotic behaviour of the iterates of the symbol. As an appendix, we deal with some particular case in the setting of weighted Banach spaces of holomorphic functions.

math.FA↗
arXiv2015

Composition operators on the Schwartz space

We study composition operators on the Schwartz space of rapidly decreasing functions. We prove that such a composition operator is never a compact operator and we obtain necessary or sufficient conditions for the range of the composition operator to be closed. These conditions are expressed in terms of multipliers for the Schwartz class and the closed range property of the corresponding operator considered in the space of smooth functions.

math.FA↗
arXiv2013

Strongly continuous semigroups on some Fréchet spaces

We prove that for a strongly continuous semigroup on the Fréchet space of all scalar sequences, its generator is a continuous linear operator and that the semigroup can be represented as exp(tA) where the exponential series converges in a very strong sense. This solves a problem posed by Conejero. Moreover, we improve recent results of Albanese, Bonet, and Ricker about semigroups on strict projective limits of Banach spaces.

math.FA↗
arXiv2013

Whitney extension operators without loss of derivatives

For a compact set, we characterize the existence of a linear extension operator E for the space of Whitney jets without loss of derivatives, that is, E satisfies the best possible continuity estimates: The supremum of all partial derivatives up to order n of E(f) is less or equal than a constant times the n-th Whitney norm of f. The characterization is a surprisingly simple purely geometric condition telling in a way that at all its points, the set is big enough in all directions.

math.FA↗