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## Publications

Publications (106)

We prove the following Farkas’ Lemma for simultaneously diagonalizable bilinear forms: If $$A_1,\ldots ,A_k$$ A 1 , … , A k , and $$B:\mathbb {R}^n \times \mathbb {R}^n \rightarrow \mathbb {R}$$ B : R n × R n → R are bilinear forms, then one—and only one—of the following holds: $$B=a_1 A_1 + \cdots + a_k A_k,$$ B = a 1 A 1 + ⋯ + a k A k , with non-...

In this survey, we provide an overview from 2008 to 2021 about the Bishop–Phelps–Bollobás theorem.KeywordsNorm attaining operatorsBishop–Phelps theoremBishop–Phelps–Bollobás property

We study the Bishop–Phelps–Bollobás property for numerical radius restricted to the case of compact operators (BPBp-nu for compact operators in short). We show that \(C_0(L)\) spaces have the BPBp-nu for compact operators for every Hausdorff topological locally compact space L. To this end, on the one hand, we provide some techniques allowing to pa...

A new estimate for the Bohr radius of the family of holomorphic functions in the n-dimensional polydisk is provided. This estimate, obtained via a new approach, is sharper than those that are known up to date.

We study the Bishop-Phelps-Bollob\'as property for numerical radius restricted to the case of compact operators (BPBp-nu for compact operators in short). We show that $C_0(L)$ spaces have the BPBp-nu for compact operators for every Hausdorff topological locally compact space $L$. To this end, on the one hand, we provide some techniques allowing to...

We introduce and investigate the mth polarization constant of a Banach space X for the numerical radius. We first show the difference between this constant and the original mth polarization constant associated with the norm by proving that the new constant is minimal if and only if X is strictly convex, and that there exists a Banach space which do...

We study the group invariant continuous polynomials on a Banach space $X$ that separate a given set $K$ in $X$ and a point $z$ outside $K$. We show that if $X$ is a real Banach space, $G$ is a compact group of $\mathcal{L} (X)$, $K$ is a $G$-invariant set in $X$, and $z$ is a point outside $K$ that can be separated from $K$ by a continuous polynomi...

Let H∞(Bc0) be the algebra of all bounded holomorphic functions on the open unit ball of c0 and M(H∞(Bc0)) the spectrum of H∞(Bc0). We prove that for any point z in the closed unit ball of ℓ∞ there exists an analytic injection of the open ball Bℓ∞ into the fiber of z in M(H∞(Bc0)), which is an isometry from the Gleason metric of Bℓ∞ to the Gleason...

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...

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

We introduce and explore a new property related to reflexivity that plays an important role in the characterization of norm attaining operators. We also present an application to the theory of compact perturbations of linear operators and characterize norm attaining scalar-valued continuous $2$-homogeneous polynomials on $\ell_{2}$.

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...

In the study of the spectra of algebras of holomorphic functions on a Banach space E , the bidual E ″ has a central role, and the spectrum is often shown to be locally homeomorphic to E ″. In this paper we consider the problem of spectra of subalgebras invariant under the action of a group (functions f such that f ○ g = f ). It is natural to attemp...

In this paper we study spaces of multiple Dirichlet series and their properties. We set the ground of the theory of multiple Dirichlet series and define the spaces H∞(C+k), k∈N, of convergent and bounded multiple Dirichlet series on C+k. We give a representation for these Banach spaces and prove that they are all isometrically isomorphic, independe...

Giving a partial answer to a conjecture formulated by Aron, Boyd, Ryan and Zalduendo, we show that if a real Banach space X is not linearly and continuously injected into a Hilbert space, then for any 2-homogeneous continuous polynomial P on X, its zero-set is not separable. For this class of spaces, we also prove that, if P is semidefinite, then i...

Given a proper holomorphic mapping \(g:\varOmega \subseteq {\mathbb {C}}^{n}\longrightarrow \varOmega ' \subseteq {\mathbb {C}}^{n}\) and an algebra of holomorphic functions \({\mathcal {B}}\) (e.g. \({\mathscr {P}}(K)\) where \(K\subset \varOmega \) is a compact set, \({\mathcal {H}}(U)\), A(U) or \({\mathcal {H}}^{\infty }(U)\) where U is an open...

The main aim of this paper is to prove a Bishop-Phelps-Bollobás type theorem on the unital uniform algebra A w∗u (B X∗ ) consisting of all w∗-uniformly continuous functions on the closed unit ball B X∗ which are holomorphic on the interior of B X∗ . We show that this result holds for A w∗u (B X∗ ) if X∗ is uniformly convex or X∗ is the uniformly co...

We study the existence of infinite-dimensional vector spaces in the sets of norm-attaining operators, multilinear forms, and polynomials. Our main result is that, for every set of permutations P of the set [1,...,n], there exists a closed infinite-dimensional Banach subspace of the space of n-linear forms on ℓ1 such that, for all nonzero elements B...

Let BX be the open unit ball of a complex Banach space X, and let H∞(BX) and Au(BX) be, respectively, the algebra of bounded holomorphic functions on BX and the subalgebra of uniformly continuous holomorphic functions on BX: In this paper we study the analytic structure of fibers in the spectrum of these two algebras. For the case of H∞ (BX), we pr...

We study the Bishop-Phelps-Bollobás property and the Bishop-Phelps-Bollobás property for numerical radius. Our main aim is to extend some known results about norm or numerical radius attaining operators to multilinear and polynomial cases. We characterize the pair $(\ell_1(X), Y)$ to have the BPBp for bilinear forms and prove that on $L_1 (\mu)$ th...

We study the Bishop-Phelps-Bollob\'as property (BPBp for short) for compact operators. We present some abstract techniques which allows to carry the BPBp for compact operators from sequence spaces to function spaces. As main applications, we prove the following results. Let $X$, $Y$ be Banach spaces. If $(c_0,Y)$ has the BPBp for compact operators,...

Given two balanced compact subsets K and L of two Banach spaces X and Y respectively such that every continuous m-homogeneous polynomial on \(X^{**}\) and on \(Y^{**}\) is approximable, for all \(m\in \mathbb {N}\), we characterize when the algebras of holomorphic germs \(\mathcal {H}(K)\) and \(\mathcal {H}(L)\) are topologically algebra isomorphi...

We exhibit a new class of Banach spaces Y such that the pair has the Bishop-Phelps-Bollobás property for operators. This class contains uniformly convex Banach spaces and spaces with the property β of Lindenstrauss. We also provide new examples of spaces in this class.

In this paper we give a new characterization of when a Banach space E has the Schur property in terms of the disk algebra. We prove that E has the Schur property if and only if .

We study the behavior of holomorphic mappings on p-compact sets in Banach spaces. We show that the image of a p-compact set by an entire mapping is a p-compact set. Some results related to the localization of p-compact sets in the predual of homogeneous polynomials are also obtained. Finally, the “size” of p-compactness of the image of the unit bal...

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_{...

Our goal is to study the Bishop-Phelps-Bollobas property for operators from c(0) into a Banach space. We first characterize those Banach spaces Y for which the Bishop-Phelps-Bollobas property holds for (l(infinity)(3), Y). Examples of spaces satisfying this condition are provided.

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...

For a sigma-finite measure mu and a Banach space Y we study the Bishop-Phelps-Bollobas property (BPBP) for bilinear forms on L-1(mu) X Y, that is, a (continuous) bilinear form on L-1(mu) X Y almost attaining its norm at (f(0), y(0)) can be approximated by bilinear forms attaining their norms at unit vectors close to (f(0), y(0)). In case that Y is...

We characterize the Banach spaces Y for which certain subspaces of operators from L1(μ)L1(μ) into Y have the Bishop–Phelps–Bollobás property in terms of a geometric property of Y, namely AHSP. This characterization applies to the spaces of compact and weakly compact operators. New examples of Banach spaces Y with AHSP are provided. We also obtain t...

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 $\mathcal{L}(X^*,Y^*)$...

We study norm attaining properties of the Arens extensions of multilinear forms defined on Banach spaces. Among other related results, we construct a multilinear form on ℓ1 with the property that only some fixed Arens extensions determined a priori attain their norms. We also study when multilinear forms can be approximated by ones with the propert...

In this survey we report on very recent results about some non-linear geometrical properties of many classes of real and complex Banach spaces and uniform algebras, including the ball algebra \(\fancyscript{A}_u(B_X)\) of all uniformly continuous functions on the closed unit ball and holomorphic on the open unit ball of a complex Banach space \(X\)...

We estimate the polynomial numerical indices of the spaces C(K) and L1(μ).

In this paper we study some geometrical properties of certain classes of uniform algebras, in particular the ball algebra image of all uniformly continuous functions on the closed unit ball and holomorphic on the open unit ball of a complex Banach space image. We prove that image has image-numerical index 1 for every image, the lushness and also th...

In this paper we provide versions of the Bishop-Phelps-Bollobás Theorem for bilinear forms. Indeed we prove the first positive result of this kind by assuming uniform convexity on the Banach spaces. A characterization of the Banach space Y satisfying a version of the Bishop-Phelps-Bollobás Theorem for bilinear forms on 1 × Y is also obtained. As a...

We study the spectra of algebras of holomorphic functions with prescribed radii of boundedness, and use these results to study the τω and τδ spectra of H(U), where U is an open subset of a non-separable Banach space. We construct τδ continuous characters on H(U) which are not evaluations at points of U. We also discuss subsets of ℓ∞ which are bound...

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...

We show that the Bishop–Phelps–Bollobás theorem holds for all bounded operators from L1(μ)L1(μ) into L∞[0,1]L∞[0,1], where μ is a σ-finite measure.

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...

We investigate certain envelopes of open sets in dual Banach spaces which are related to extending holomorphic functions. We give a variety of examples of absolutely convex sets showing that the extension is in many cases not possible. We also establish connections to the study of iterated weak* sequential closures of convex sets in the dual of sep...

In this article we study the interplay of the theory of classical Dirichlet series in one complex variable with recent development on monomial expansions of holomorphic functions in infinitely many variables. For a given Dirichlet series we obtain new strips of convergence in the complex plane related to Bohr’s classical strips of uniform but non a...

The aim of this paper is to show that there exist infinite dimensional Banach spaces of functions that, except for 0, satisfy properties that apparently should be destroyed by the linear combination of two of them. Three of these spaces are: a Banach space of differentiable functions on ℝn failing the Denjoy-Clarkson property; a Banach space of non...

If X is an Asplund space, then every uniformly continuous function on BX* which is holomorphic on the open unit ball, can be perturbed by a w* continuous and homogeneous polynomial on X* to obtain a norm attaining function on the dual unit ball. This is a consequence of a version of Bourgain-Stegall’s variational
principle. We also show that the se...

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...

Let H be a (real or complex) Hilbert space. Using spectral theory and properties of the Schatten–Von Neumann operators, we prove
that every symmetric tensor of unit norm in H [^(Ä)] s,psH{H \hat{\otimes} _{s,\pi _{s}}H} is an infinite absolute convex combination of points of the form xÄx{x\otimes x} with x in the unit sphere of the Hilbert space. W...

We study two-dimensional Banach spaces with polynomial numerical indices equal to zero.

We study two-dimensional Banach spaces with polynomial numerical indices equal to zero.

Let X be a separable Banach space. We provide an explicit construction of a sequence in X that tends to 1 in norm but which is weakly dense. Our interest in the result stated in the Abstract was motivated by two theorems. First, in their work on hypercyclic operators, K. Chan and R. Sanders proved the following: Theorem 1. (Chan and Sanders (3)) Fo...

Given an entire mapping $f\in \mathcal{H}_b(X,X)$ of bounded type from a Banach space $X$ into $X$, we denote by $\overline{f}$ the Aron-Berner extension of $f$ to the bidual $X^{\ast\ast}$ of $X$. We show that $\overline{g\circ f} = \overline{g}\circ \overline{f}$ for all $f, g\in \mathcal{H}_b(X,X)$ if $X$ is symmetrically regular. We also give a...

We prove the Bishop–Phelps–Bollobás theorem for operators from an arbitrary Banach space X into a Banach space Y whenever the range space has property β of Lindenstrauss. We also characterize those Banach spaces Y for which the Bishop–Phelps–Bollobás theorem holds for operators from ℓ1ℓ1 into Y. Several examples of classes of such spaces are provid...

We study the relation between the polynomial numerical indices of a complex vector-valued function space and the ones of its range space. It is proved that the spaces C ( K , X ) and L ∞ (μ, X ) have the same polynomial numerical index as the complex Banach space X for every compact Hausdorff space K and every σ-finite measure μ, which does not hol...

Bohr showed that the width of the strip (in the complex plane) on which a given Dirichlet series
åan/ ns, s Î \mathbbC{\sum a_n/ n^s, \, s \in \mathbb{C}}, converges uniformly but not absolutely, is at most 1/2, and Bohnenblust-Hille that this bound in general is optimal. We
prove that for a given infinite dimensional Banach space Y the width of...

We prove a multilinear version of Phelps' Lemma: if the zero sets of multilinear forms of norm one are 'close', then so are the multilinear forms.

Let X be a Banach space which has an unconditional basis and is a subspace of some ℒ1-space Y. We show that X = ℓ1 if and only if every m-linear form T on X, m ∈ N, has an m-linear extension T̃ to Y satisfying ||T̃|| ≤ Cm||T||, where C > 0 is a constant independent of m. If we replace m-linear forms by m-homogeneous polynomials, then we can only sh...

We study when the Daugavet equation is satisfied for weakly compact polynomials on a Banach space X, i.e. when the equality ||Id + P|| = 1 + ||P|| is satisfied for all weakly compact polynomials P : X → X. We show that this is the case when X = C(K), the real or complex space of continuous functions on a compact space K without isolated points. We...

If $\Omega$ is a non empty convex open subset of a Silva space $E$ with the origin in its boundary and $F$ is a Fréchet space, we study the spaces of holomorphic mappings from $\Omega$ into $F$ such that its differentials can be continuously extended to the origin. These spaces are compared with the spaces of holomorphic mappings with asymptotic ex...

We show that the set of N-linear mappings on a product of N Banach spaces such that all their Arens extensions attain their norms (at the same element) is norm dense in the space of all bounded N-linear mappings.

In this paper, we introduce the polynomial numerical index of order $k$ of a Banach space, generalizing to $k$-homogeneous polynomials the ‘classical’ numerical index defined by Lumer in the 1970s for linear operators. We also prove some results. Let $k$ be a positive integer. We then have the following:
(i) $n^{(k)}(C(K))=1$ for every scattered co...

Let U and V be convex and balanced open subsets of the Banach spaces X and Y respectively. In this paper we study the following question: Given two Fréchet algebras of holomorphic functions of bounded type on U and V respectively that are algebra-isomorphic, can we deduce that X and Y (or X * and Y *) are isomor-phic? We prove that if X * or Y * ha...

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.

We show that on a complex Banach space X, the functions uniformly continuous on the closed unit ball and holomorphic on the open unit ball that attain their norms are dense provided that X has the Radon–Nikodym property. We also show that the same result holds for Banach spaces having a strengthened version of the approximation property but conside...

Let be the Banach space of all complex-valued continuous functions on a compact Hausdorff space K. We study when the following statement holds: every norm attaining n-homogeneous complex polynomial on attains its norm at extreme points. We prove that this property is true whenever K is a compact Hausdorff space of dimension less than or equal to on...

We obtain general lower and upper estimates for the first and the second Bohr radii of bounded complete Reinhardt domains in Cn.

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...

Let ∑|α|=m sαzα, z ∈ ℂn be a unimodular m-homogeneous polynomial in n variables (i.e. |sα| = 1 for all multi indices α), and let R ⊂ ℂn be a (bounded complete) Reinhardt domain. We give lower bounds for the maximum modulus supz ∈ R | ∑|α: m sαzα |, and upper estimates for the average of these maximum moduli taken over all possible m-homogeneous Ber...

We show that for every Banach space $X$ the set of 2-homogeneous continuous polynomials whose canonical extension to $X^{\ast\ast}$ attain their norm is a dense subset of the space of all 2-homogeneous continuous polynomials $\mathcal{P}(^2X)$.

In 1914 Harald Bohr published the following surprising result: Suppose that \Sigma(k=0)(proportional to)a(k)z(k)\ less than or equal to 1 for each z in the open unit disk. Then Sigma(k=0)(proportional to)\a(k)z(k)\ less than or equal to 1when \z\ < 1/3, and moreover the radius 1/3 is best possible. Recently several authors studied Bohr's power theo...

In 1914 Harald Bohr published the following surprising result: Suppose that |∑K=0∞ akzk| ≤ 1 for each z in the open unit disk. Then ∑K=0∞ |akzk| ≤ 1 when |z| < 1/3, and moreover the radius 1/3 is best possible. Recently several authors studied Bohr's power theorem in higher dimension: Given a Banach space X = (ℂn, ∥.∥), what is the largest radius K...

It is shown that for every quasi-normed ideal Q{\cal Q} of
n-homogeneous continuous polynomials between
Banach spaces there is a quasi-normed ideal A{\cal A} of
n-linear continuous mappings A{\cal A} such that
q Î Qq \in {\cal Q} if and only if the associated n-linear
mapping
\checkq\check{q} of q is in A{\cal A}.

Our main result shows that every Montel Köthe echelon or coechelon space E of order 1 < p ≤ ∞ is nuclear if and only if for every (some) m ≥ 2 the space ((mE), τ0) of m-homegeneus polynomials on E endowed with the compact-open topology τ0 has an unconditional basis if and only if the space (ℋ(E), τδ) of holomorphic functions on E endowed with the b...

No infinite dimensional Banach space X is known which has the property that for m⩾2 the Banach space of all continuous m-homogeneous polynomials on X has an unconditional basis. Following a program originally initiated by Gordon and Lewis we study unconditionality in spaces of m-homogeneous polynomials and symmetric tensor products of order m in Ba...

Estudiamos algunas situaciones donde encontramos un problema que, a primera vista, parece no tener solución. Pero, de hecho, existe un subespacio vectorial grande de soluciones del mismo. We study some situations where one encounters a problem which, atrst glance, appears to have no solutions at all. But, actually, there is a large linear subspace...

A Banach space E is said to be (symmetrically) regular if every continuous (symmetric) linear mapping from E to E′ is weakly compact. For a complex Banach space E and a complex Banach algebra F, let b(E, F) denote the algebra of holomorphic mappings from E to F which are bounded on bounded sets. We endow b(E, F) with the usual Fréchet topology. (b(...

For every open subset G of $$ \mathbb{C}^N $$ and for every continuous, strictly positive weight v on G, the Banach space of all the holomorphic functions f on G such that $$ v|f| $$ vanishes at infinity on G, endowed with the natural weighted sup-norm, is isomorphic to a closed subspace of the Banach space c 0; hence it is reflexive if and only if...

LetH(U) denote the space of all holomorphic functions on an open subsetUof a separable Fréchet spaceE. Let τωdenote the compact-ported topology onH(U) introduced by Nachbin. LetG(U) denote the predual ofH(U) constructed by Mazet. In our main result we show thatEis quasi-normable if and only ifG(U) is quasi-normable if and only if (H(U), τω) satisfi...

A Banach space E is known to be Arens regular if every continuous linear mapping from E to E′ is weakly compact. Let U be an open subset of E, and let Hb(U) denote the algebra of analytic functions on U which are bounded on bounded subsets of U lying at a positive distance from the boundary of U. We endow Hb(U) with the usual Fréchet topology.Mb(U)...

In [13] Mazet proved the following result.
If U is an open subset of a locally convex space E then there exists a complete locally convex space ( U ) and a holomorphic mapping δ U : U → ( U ) such that for any complete locally convex space F and any f ɛ ℋ ( U;F ), the space of holomorphic mappings from U to F , there exists a unique linear mapping...

By following an idea of O. Nicodemi [Lect. Notes Math. 843, 534-546 (1981; Zbl 0482.46012)], we study certain sequences of extension operators for multilinear mappings on Banach spaces starting from any given extension operator for linear mappings. In this way, we obtain several new properties of the extension operators previously studied by Aron,...

We show that holomorphic mappings of bounded type defined on Fréchet spaces extend to the bidual. The relationship between holomorphic mappings of bounded type and of uniformly bounded type is discussed and some algebraic and topological properties of the space of all entire mappings of (uniformly) bounded type are proved, for example a holomorphic...

We study the holomorphic functions of bounded type defined on (DF)-spaces. We prove that they are of uniformly bounded type. The space of all these functions is a Fréchet space with its natural topology. Some consequences and related results are obtained.

For a Banach space E, we prove that the Fréchet space b(E) is the strong dual of an (LB)-space, b(E), which leads to a linearization of the holomorphic mappings of bounded type. It is also shown that the holomorphic functions defined on (DFC)-spaces are of uniformly bounded type.

We study topological properties of the space (ℋ(E),τ δ ) for a fully nuclear space E with basis. This leads us to a negative solution of a problem of Mujica and Nachbin concerning the relationship between properties of holomorphic mappings defined on a locally convex space E and linear properties of a predual of ℋ(E). The solution is provided by th...

Several results and examples about locally bounded sets of holomorphic mappings defined on certain classes of locally convex spaces (Baire spaces, $(DF)$-spaces, $C(X)$-spaces) are presented. Their relation with the classification of locally convex spaces according to holomorphic analogues of barrelled and bornological properties of the linear theo...