Pablo Sevilla-Peris

• Polytechnic University of Valencia

In this chapter, an introduction to the theory of transitive and mean ergodic operators on locally convex spaces is given. The relation of transitive operators and hypercyclic and chaotic operators is analysed. Criteria, due to Godefroy and Shapiro and to Bès and Peris, to decide whether or not an operator is transitive are included. Theorems due t...

This chapter investigates pointwise and uniform convergence of sequences and series of functions including many examples. Once the main results and examples are given, we look at pointwise and uniform convergence for different relevant situations: power series, Fourier series, and Dirichlet series. We present the radius of convergence of power seri...

In this chapter, spaces of holomorphic functions and of infinitely differentiable functions are defined and studied. Their locally convex structure is analysed. The theorems of Montel and Vitali are proved. For the space of holomorphic functions on the disc and on the complex plane, a representation as a sequence space is given. Two descriptions of...

This chapter starts recalling the definitions of Hausdorff topological space, metric space, and normed space. Examples of Banach sequence spaces, of continuous functions and of measurable functions, are given. Seminorms are introduced and they are used to introduce the concept of Hausdorff locally convex space. Bounded sets are defined and their pr...

The aim of this chapter is to present the duality theory and to prove the fundamental theorems of the theory of locally convex spaces in a direct self-contained way. Hyperplanes are defined and the analytic proof of Hahn–Banach theorem is explained with details. The separation theorems of convex sets are obtained as a consequence. Precompact subset...

In this chapter, an introduction to the theory of transitive and mean ergodic operators on locally convex spaces is given. The relation of transitive operators and hypercyclic and chaotic operators is analysed. Criteria, due to Godefroy and Shapiro and to Bès and Peris, to decide whether or not an operator is transitive are included. Theorems due t...

We give a version of the Montel theorem for Hardy spaces of holomorphic functions on an infinite dimensional space. Precisely, we show that any bounded sequence of holomorphic functions in some Hardy space, has a subsequence that converges uniformly over compact subsets to a function that also belongs to the same Hardy space. As a by-product of our...

Decoupling inequalities disentangle complex dependence structures of random objects so that they can be analyzed by means of standard tools from the theory of independent random variables. We study decoupling inequalities for vector-valued homogeneous polynomials evaluated at random variables. We focus on providing geometric conditions ensuring dec...

We characterize the space of multipliers from the Hardy space of Dirichlet series $\mathcal H_p$ into $\mathcal H_q$ for every $1 \leq p,q \leq \infty$. For a fixed Dirichlet series, we also investigate some structural properties of its associated multiplication operator. In particular, we study the norm, the essential norm, and the spectrum for an...

We study the Hardy space of translated Dirichlet series \({\mathscr {H}}_{+}\). It consists on those Dirichlet series \(\sum a_n n^{-s}\) such that for some (equivalently, every) \(1 \le p < \infty \), the translation \(\sum {a_{n}}n^{-(s+\frac{1}{\sigma })}\) belongs to the Hardy space \({\mathscr {H}}^{p}\) for every \(\sigma >0\). We prove that...

Inspired by a recent article on Fréchet spaces of ordinary Dirichlet series \(\sum a_n n^{-s}\) due to J. Bonet, we study topological and geometrical properties of certain scales of Fréchet spaces of general Dirichlet spaces \(\sum a_n e^{-\lambda _n s}\) focus on the Fréchet space of \(\lambda \)-Dirichlet series \(\sum a_n e^{-\lambda _n s}\) whi...

We study mean ergodic composition operators on infinite dimensional spaces of holomorphic functions of different types when defined on the unit ball of a Banach or a Hilbert space: that of all holomorphic functions, that of holomorphic functions of bounded type and that of bounded holomorphic functions. Several examples in the different settings ar...

Decoupling inequalities disentangle complex dependence structures of random objects so that they can be analyzed by means of standard tools from the theory of independent random variables. We study decoupling inequalities for vector-valued homogeneous polynomials evaluated at random variables. We focus on providing geometric conditions ensuring dec...

We study mean ergodic composition operators on infinite dimensional spaces of holomorphic functions of different types when defined on the unit ball of a Banach or a Hilbert space: that of all holomorphic functions, that of holomorphic functions of bounded type and that of bounded holomorphic functions. Several examples in the different settings ar...

Inspired by a recent article on Fr\'echet spaces of ordinary Dirichlet series $\sum a_n n^{-s}$ due to J.~Bonet, we study topological and geometrical properties of certain scales of Fr\'echet spaces of general Dirichlet spaces $\sum a_n e^{-\lambda_n s}$. More precisely, fixing a frequency $\lambda = (\lambda_n)$, we focus on the Fr\'echet space of...

We give a version of the Montel theorem for Hardy spaces of holomorphic functions on an infinite dimensional space. As a by-product, we provide a Montel-type theorem for the Hardy space of Dirichlet series. This approach also gives an elementary proof of Montel theorem for the classical one-variable Hardy spaces.

We study Hausdorff–Young-type inequalities for vector-valued Dirichlet series which allow us to compare the norm of a Dirichlet series in the Hardy space H p ( X ) \mathcal {H}_{p} (X) with the q q -norm of its coefficients. In order to obtain inequalities completely analogous to the scalar case, a Banach space must satisfy the restrictive notion o...

We study the Hardy space of translated Dirichlet series $\mathcal{H}_{+}$. It consists on those Dirichlet series $\sum a_n n^{-s}$ such that for some (equivalently, every) $1 \leq p < \infty$, the translation $\sum{a_{n}}n^{-(s+\frac{1}{\sigma})}$ belongs to the Hardy space $\mathcal{H}^{p}$ for every $\sigma>0$. We prove that this set, endowed wit...

We study composition operators on spaces of double Dirichlet series, focusing our interest on the characterization of the composition operators of the space of bounded double Dirichlet series \({\mathcal {H}}^\infty ({\mathbb {C}}_+^2)\). We also show how the composition operators of this space of Dirichlet series are related to the composition ope...

We study some dynamical properties of composition operators defined on the space P(Xm) of m-homogeneous polynomials on a Banach space X when P(Xm) is endowed with two different topologies: the one of uniform convergence on compact sets and the one defined by the usual norm. The situation is quite different for both topologies: while in the case of...

We study some dynamical properties of composition operators defined on the space $\mathcal{P}(^m X)$ of $m$-homogeneous polynomials on a Banach space $X$ when $\mathcal{P}(^m X)$ is endowed with two different topologies: the one of uniform convergence on compact sets and the one defined by the usual norm. The situation is quite different for both t...

Cambridge Core - Abstract Analysis - Dirichlet Series and Holomorphic Functions in High Dimensions - by Andreas Defant

We give a self-contained treatment of symmetric Banach sequence spaces and some of their natural properties. We are particularly interested in the symmetry of the norm and the existence of symmetric linear functionals. Many of the presented results are known or commonly accepted but are not found in the literature.

For $1 < r \le 2$, we study the set of monomial convergence for spaces of holomorphic functions over $\ell_r$. For $ H_b(\ell_r)$, the space of entire functions of bounded type in $\ell_r$, we prove that $\mbox{mon} H_b(\ell_r)$ is exactly the Marcinkiewicz sequence space $m_{\Psi_r}$ where the symbol $\Psi_r$ is given by $\Psi_r(n) := \log(n + 1)^...

We study Hausdorff-Young inequalities for vector-valued Dirichlet series. These are inequalities that relate some norm of the coefficients $(a_{n})_n$ and the norm of the Dirichlet series in the Hardy space $\mathcal{H}_{p} (X)$. This leads us in a natural way to consider different type/cotype properties of the space. Restrictive properties as Four...

We study composition operators on spaces of double Dirichlet series, focusing our interest on the characterization of the composition operators of the space of bounded double Dirichlet series $\HCdos$. We also show how the composition operators of this space of Dirichlet series are related to the composition operators of the corresponding spaces of...

We present an abstract approach to the abscissas of convergence of vector-valued Dirichlet series. As a consequence we deduce that the abscissas for Hardy spaces of Dirichlet series are all equal. We also introduce and study weak versions of the abscissas for scalar-valued Dirichlet series.

We present an abstract approach to the abscissas of convergence of vector-valued Dirichlet series. As a consequence we deduce that the abscissas for Hardy spaces of Dirichlet series are all equal. We also introduce and study weak versions of the abscissas for scalar-valued Dirichlet series.

While the article was in publication process, we found a mistake in a key tool for the proof one of the main results. As a consequence, our result on the ball Au(BX) algebra remains open. For the algebra Hb(X) we obtain a weaker statement, which still extends previous work in the subject. In this note we enumerate those results which should be omit...

The Cluster Value Theorem is known for being a weak version of the classical Corona Theorem. Given a Banach space $X$, we study the Cluster Value Problem for the ball algebra $A_u(B_X)$, the Banach algebra of all uniformly continuous holomorphic functions on the unit ball $B_X$; and also for the Fr\'echet algebra $H_b(X)$ of holomorphic functions o...

Let $\mathcal{H}_\infty$ be the set of all ordinary Dirichlet series $D=\sum_n a_n n^{-s}$ representing bounded holomorphic functions on the right half plane. A multiplicative sequence $(b_n)$ of complex numbers is said to be an $\ell_1$-multiplier for $\mathcal{H}_\infty$ whenever $\sum_n |a_n b_n| < \infty$ for every $D \in \mathcal{H}_\infty$. W...

We employ a classical result by Toeplitz (1913) and the seminal work by Bohnenblust and Hille on Dirichlet series (1931) to show that the set of continuous m-homogeneous non-analytic polynomials on c0 contains an isomorphic copy of ℓ1. Moreover, we can have this copy of ℓ1 in such a way that every non-zero element of it fails to be analytic at prec...

We introduce non-linear versions of the classical cotype of Banach spaces. We show that spaces with l.u.st and cotype, and that spaces having Fourier cotype enjoy our non-linear cotype. We apply these concepts to get results on convergence of vector-valued power series in infinite many variables and on $\ell_{1}$-multipliers of vector-valued Dirich...

Let χ(m,n) be the unconditional basis constant of the monomial basis zα, α ∈ ℕno with |α| = m, of the Banach space of all m-homogeneous polynomials in n complex variables, endowed with the supremum norm on the n-dimensional unit poly disc Dn. We prove that the quotient of supm nm√supmχ(m,n) and √n/logn tends to 1 as n → ∞. This reflects a quite pre...

We investigate the summability of the coeffcients of m-homogeneou polynomials and m-linear mappings defined on ℓp spaces. In our research we obtai results on the summability of the coeffcients of m-linear mappings defined on ℓp1 Xℓpm. The rst results in this respect go back to Littlewood [17] and Bohnenblus and Hille [6] for bilinear and m-linear f...

We give a variant of the Bohenblust-Hille inequality which, for certain families of polynomials, leads to constants with polynomial growth in the degree.

Denote by $\Omega(n)$ the number of prime divisors of $n \in \mathbb{N}$ (counted with multiplicities). For $x\in \mathbb{N}$ define the Dirichlet-Bohr radius $L(x)$ to be the best $r>0$ such that for every finite Dirichlet polynomial $\sum_{n \leq x} a_n n^{-s}$ we have $$ \sum_{n \leq x} |a_n| r^{\Omega(n)} \leq \sup_{t\in \mathbb{R}} \big|\sum_{...

Hartman proved in 1939 that the width of the largest possible strip in the complex plane, on which a Dirichlet series $\sum_n a_n n^{-s}$ is uniformly a.s.-sign convergent (i.e., $\sum_n \varepsilon_n a_n n^{-s}$ converges uniformly for almost all sequences of signs $\varepsilon_n =\pm 1$) but does not convergent absolutely, equals $1/2$. We study...

By the von Neumann inequality for homogeneous polynomials there exists a positive constant $C_{k,q}(n)$ such that for every $k$-homogeneous polynomial $p$ in $n$ variables and every $n$-tuple of commuting operators $(T_1, \dots, T_n)$ with $\sum_{i=1}^{n} \Vert T_{i} \Vert^{q} \leq 1$ we have \[ \|p(T_1, \dots, T_n)\|_{\mathcal L(\mathcal H)} \leq...

Denote by Ω(n) the number of prime divisors of n ∈ N (counted with multi-plicities). For x ∈ N define the Dirichlet-Bohr radius L(x) to be the best r > 0 such that for every finite Dirichlet polynomial n≤x a n n −s we have n≤x |a n |r Ω(n) ≤ sup t ∈R n≤x a n n −i t. We prove that the asymptotically correct order of L(x) is (log x) 1/4 x −1/8. Follo...

We estimate the \(\ell _1\) -norm \(\sum _{n=1}^N \Vert a_n\Vert \) of finite Dirichlet polynomials \(\sum _{n=1}^N a_n n^{-s},\,s \in {\mathbb {C}}\) with coefficients \(a_n\) in a Banach space. Our estimates quantify several recent results on Bohr’s strips of uniform but non absolute convergence of Dirichlet series in Banach spaces.

For two complex Banach spaces $X$ and $Y$, in this paper we study the generalized spectrum $\mathcal{M}_b(X,Y)$ of all non-zero algebra homomorphisms from $\mathcal{H}_b(X)$, the algebra of all bounded type entire functions on $X$, into $\mathcal{H}_b(Y)$. We endow $\mathcal{M}_b(X,Y)$ with a structure of Riemann domain over $\mathcal{L}(X^*,Y^*)$...

The Bohr-Bohnenblust-Hille theorem states that the width of the strip in the complex plane on which an ordinary Dirichlet series ∑ n a n n -s converges uniformly but not absolutely is less than or equal to 1 2, and this estimate is optimal. Equivalently, the supremum of the absolute convergence abscissas of all Dirichlet series in the Hardy space ℋ...

In 1931 H. F. Bohnenblust and E. Hille [Ann. Math. (2) 32, 600–622 (1931; Zbl 0001.26901)] published a very important paper in which not only they did solve a long standing problem on convergence of Dirichlet series, but also gave a general version of a celebrated inequality of Littlewood. Although it is full of extremely valuable mathematical idea...

We present a brief and informal account on the so called Bohr's absolute convergence problem on Dirichlet series, from its statement and solution in the beginings of the 20th century to some of its recent variations.

We investigate the summability of the coefficients of $m$-homogeneous polynomials and $m$-linear mappings defined on $\ell_{p}$-spaces. In our research we obtain results on the summability of the coefficients of $m$-linear mappings defined on $\ell_{p_{1}} \times \cdots \times \ell_{p_{m}}$. The first results in this respect go back to Littlewood a...

We study extendibility of diagonal multilinear operators from ℓp to ℓq spaces. We determine the values of p and q for which every diagonal n -linear operator is extendible, and those for which the only extendible ones are integral. We address the same question for multilinear forms on ℓp .

We study Hahn-Banach extensions of multilinear forms defined on Banach sequence spaces. We characterize $c_0$ in terms of extension of bilinear forms, and describe the Banach sequence spaces in which every bilinear form admits extensions to any superspace.

If E is a Banach sequence space, then each holomorphic function defines a formal power series ∑α cα(f) zα. The problem of when such an expansion converges absolutely and actually represents the function goes back to the very beginning of the theory of holomorphic functions on infinite-dimensional spaces. Several very deep results have been given fo...

Each bounded holomorphic function on the infinite dimensional polydisk $\mathbb{D}^\infty$, $f \in H_\infty(\mathbb{D}^\infty)$, defines a formal monomial series expansion that in general does not converge to $f$. The set $\mon H_\infty(\mathbb{D}^\infty)$ contains all $ z $'s in which the monomial series expansion of each function $f \in H_\infty(...

We survey the most relevant recent developments on the research of the spectra of algebras of analytic functions. We concentrate mainly on three algebras, the Banach algebra H ∞ (B) of all bounded holomorphic functions on the unit ball B of a complex Banach space X, the Banach algebra of the ball A u (B), and the Fréchet algebra H b (X) of all enti...

Each Dirichlet series $D = \sum_{n=1}^{\infty} a_n \frac{1}{n^s}$, with variable $s \in \mathbb{C}$ and coefficients $a_n \in \mathbb{C}$, has a so called Bohr strip, the largest strip in $\mathbb{C}$ on which $D$ converges absolutely but not uniformly. The classical Bohr-Bohnenblust-Hille theorem states that the width of the largest possible Bohr...

To the memory of Goyo Sevilla, a good, honest man. Abstract. In this paper we give general conditions on a countable family V of weights on an unbounded open set U in a complex Banach space X such that the weighted space HV (U) of holomorphic functions on U has a Fréchet algebra structure. For that kind of weights it is shown that the spectrum of H...

We unify Littlewood's classical 4/3-inequality (a forerunner of Grothendieck's inequality) together with its m-linear extension due to Bohnenblust and Hille (which originally settled Bohr's absolute convergence problem for Dirichlet series) with a scale of inequalties of Bennett and Carl in ℓp-spaces (which are of fundamental importance in the theo...

We consider weighted algebras of holomorphic functions on a Banach space. We determine conditions on a family of weights that assure that the corresponding weighted space is an algebra or has polynomial Schauder decompositions. We study the spectra of weighted algebras and endow them with an analytic structure. We also deal with composition operato...

We establish Hölder-type inequalities for Lorentz sequence spaces and their duals. In order to achieve these and some related inequalities, we study diagonal multilinear forms in general sequence spaces, and obtain estimates for their norms. We also consider norms of multilinear forms in different Banach multilinear ideals.

We establish H\"older type inequalities for Lorentz sequence spaces and their duals. In order to achieve these and some related inequalities, we study diagonal multilinear forms in general sequence spaces, and obtain estimates for their norms. We also consider norms of multilinear forms in different Banach multilinear ideals.

We establish H\"older type inequalities for Lorentz sequence spaces and their duals. In order to achieve these and some related inequalities, we study diagonal multilinear forms in general sequence spaces, and obtain estimates for their norms. We also consider norms of multilinear forms in different Banach multilinear ideals.

A general theory of limit orders for ideals of multilinear forms is developed. We relate the limit order of an ideal to those of its maximal hull and its adjoint ideal. We study the limit orders of the ideals of dominated and multiple summing multilinear forms. Finally, estimates of the diagonal of a (non-necessarily diagonal) multilinear form are...

In this paper discrete approximate processes of mixed diffusion problems under spatial uncertainty are constructed using random difference schemes. Conditions for the existence of a series stochastic solution process are given using a stochastic separation of variable method. Expectation and standard deviation of the approximate stochastic processe...

Since the concept of limit order is a useful tool to study operator ideals, we propose an analogous definition for ideals of multilinear forms. From the limit orders of some special ideals (of nuclear, integral, r-dominated and extendible multilinear forms) we derive some properties of them and show differences between the bilinear and n-linear cas...

We characterise continuity of composition operators on weighted spaces of holomorphic functions H v (B X ), where B X is the open unit ball of a Banach space which is homogeneous, that is, a JB *-triple.

This paper deals with the construction of approximating processes for solving mixed problems related to the diffusion equation where the initial condition is assumed to be a stochastic process and the diffusion coefficient is a random variable. Expectation and standard deviation of the approximation processes are given.

Using the Mellin transform a new method for solving the Black—Scholes equation is proposed. Our approach does not require either variable transformations or solving diffusion equations.

In this paper a new method for solving Black–Scholes equation is proposed. The approach is based on the Mellin transform. A numerical procedure for the approximation of the solution is given.

This paper deals with the construction of approximate numerical pro-cesses of mixed diffusion models under spatial uncertainty in the diffusion coefficient and the source term. After discretization, the stochastic discrete problem is solved using a stochastic separation of the variables method.

Our aim in this paper is to study weak compactness of composition operators between weighted spaces of holomorphic functions on the unit ball of a Banach space.

In this paper we study composition operators between weighted spaces of holo-morphic functions defined on the open unit ball of a Banach space. Necessary and sufficient conditions are given for composition operators to be compact. We show that new phenomena appear in the infinite-dimensional setting different from the ones of the finite-dimensional...

We give asymptotically correct estimations for the cotype 2 constant C2(P(mX n )) ofthe spaceP(mX n ) of allm-homogenous polynomials onX n , the span of the firstn sequencese k =(\gdkj )j in a Banach sequence spaceX. Applications to Minkowski, Orlicz and Lorentz sequence spaces are given.

We consider, using various tensor norms, the completed tensor product of two unital lmc algebras one of which is commutative. Our main result shows that when the tensor product of two Q-algebras is an lmc algebra, then it is a Q-algebra if and only if pointwise invertibility implies invertibility (as in the Gelfand theory). This is always the case...

RESUMEN La Tesis Sobre espacios y ´algebras de funciones holomorfas se estructura en tres cap´ýtulos diferentes. En cada uno de ellos se aborda un problema diferente. El primer cap´ýtulo se dedica al estudio de los operadores de composici´on. La idea original es bastante sencilla y natural. Tomamos el disco unidad complejo, que denotamos D, y una f...