arXiv subjects

Daniel Galicer

Publications and source records attributed to Daniel Galicer.

At least 37 records · Page 2Linked to original sources

arXiv2022

Multipliers for Hardy spaces of Dirichlet series

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 operator of this kind. We exploit the existing natural identification of spaces of Dirichlet series with spaces of holomorphic functions in infinitely many variables and apply several methods from complex and harmonic analysis to obtain our results. As a byproduct we get analogous statements on such Hardy spaces of holomorphic functions.

math.CV↗

arXiv2021

Continuous quantitative Helly-type results

Brazitikos' results on quantititative Helly-type theorems (for the volume and for the diameter) rely on the work of Srivastava on sparsification of John's decompositions. We change this technique by a stronger recent result due to Friedland and Youssef. This, together with an appropriate selection in the accuracy of the approximation, allow us to obtain Helly-type versions which are sensitive to the number of convex sets involved.

math.MG↗

arXiv2021

Hardy space of translated Dirichlet series

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 $\sum{a_{n}}n^{-(s+\frac{1}σ)}$ belongs to the Hardy space $\mathcal{H}^{p}$ for every $σ>0$. We prove that this set, endowed with the topology induced by the seminorms $\left\{\Vert\cdot\Vert_{2,k}\right\}_{k\in\mathbb{N}}$ (where $\Vert\sum{a_{n}n^{-s}}\Vert_{2,k}$ is defined as $\big\Vert\sum{a_n n^{-(s+\frac{1}{k})}} \big\Vert_{\mathcal{H}^{2}}$), is a Fréchet space which is Schwartz and non nuclear. Moreover, the Dirichlet monomials $\{n^{-s}\}_{n \in \mathbb N}$ are an unconditional Schauder basis of $\mathcal H_+$. In the spirit of Gordon and Hedenmalm's work, we completely characterize the composition operator on the Hardy space of translated Dirichlet series. Moreover, we study the superposition operators on $\mathcal{H}_{+}$ and show that every polynomial defines an operator of this kind. We present certain sufficient conditions on the coefficients of an entire function to define a superposition operator. Relying on number theory techniques we exhibit some examples which do not provide superposition operators. We finally look at the action of the differentiation and integration operators on these spaces.

math.FA↗

arXiv2020

The polarization constant of finite dimensional complex spaces is one

The polarization constant of a Banach space $X$ is defined as $$\mathbf c(X):= \limsup\limits_{k\rightarrow \infty} \mathbf c(k, X)^\frac{1}{k},$$ where $\mathbf c(k, X)$ stands for the best constant $C>0$ such that $ \Vert \overset{\vee}{P} \Vert \leq C \Vert P \Vert$ for every $k$-homogeneous polynomial $P \in \mathcal P(^kX)$. We show that if $X$ is a finite dimensional complex space then $\mathbf c(X)=1$. We derive some consequences of this fact regarding the convergence of analytic functions on such spaces.The result is no longer true in the real setting. Here we relate this constant with the so-called Bochnak's complexification procedure. We also study some other properties connected with polarization. Namely, we provide necessary conditions related with the geometry of $X$ for $\mathbf c(2,X)=1$ to hold. Additionally we link polarization's constants with certain estimates of the nuclear norm of the product of polynomials.

math.FA↗

arXiv2020

Impossibility of memory in hidden-signaling models for quantum correlations

We consider a toy model for non-local quantum correlations in which nature resorts to some form of hidden signaling (i.e., signaling between boxes but not available to the users) to generate correlations. We show that if such a model also had memory, the parties would be able to exploit the hidden-signaling and use it to send a message, achieving faster-than-light communication. Given that memory is a resource easily available for any physical system, our results add evidence against hidden signaling as the mechanism behind nature's non-local behavior

quant-ph↗

arXiv2020

Characterization of non-signaling correlations from mutual information

We present a characterization of the set of non-signaling correlations in terms of a two dimensional representation that involves the maximal value of a Bell functional and the mutual information between the parties. In particular, we apply this representation to the bipartite Bell scenario with two measurements and two outcomes. In terms of these physically meaningful quantities and through numerical optimization methods and some analytical results, we investigate the frontier between the different subsets of the non-signaling correlations, focussing on the quantum and post-quantum ones. Finally, we show that the Tsirelson bound appears as a singular point in this context without resorting to quantum mechanics.

quant-ph↗

arXiv2020

A Montel-type theorem for Hardy spaces of holomorphic functions

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.

math.FA↗

arXiv2020

Asymptotic estimates for the largest volume ratio of a convex body

The largest volume ratio of given convex body $K \subset \mathbb{R}^n$ is defined as $$\mbox{lvr}(K):= \sup_{L \subset \mathbb{R}^n} \mbox{vr}(K,L),$$ where the $\sup$ runs over all the convex bodies $L$. We prove the following sharp lower bound $$c \sqrt{n} \leq \mbox{lvr}(K),$$ for every body $K$ (where $c>0$ is an absolute constant). This result improves the former best known lower bound, of order $\sqrt{\frac{n}{\log \log(n)}}$. We also study the exact asymptotic behavior of the largest volume ratio for some natural classes. In particular, we show that $\mbox{lvr}(K)$ behaves as the square root of the dimension of the ambient space in the following cases: if $K$ is the unit ball of an unitary invariant norm in $\mathbb{R}^{d \times d}$ (e.g., the unit ball of the $p$-Schatten class $S_p^d$ for any $1 \leq p \leq \infty$), $K$ is the the unit ball of the full/symmetric tensor product of $\ell_p$-spaces endowed with the projective or injective norm or $K$ is unconditional.

math.MG↗

arXiv2019

The minimal volume of simplices containing a convex body

Let $K \subset \mathbb R^n$ be a convex body with barycenter at the origin. We show there is a simplex $S \subset K$ having also barycenter at the origin such that $\left(\frac{vol(S)}{vol(K)}\right)^{1/n} \geq \frac{c}{\sqrt{n}},$ where $c>0$ is an absolute constant. This is achieved using stochastic geometric techniques. Precisely, if $K$ is in isotropic position, we present a method to find centered simplices verifying the above bound that works with very high probability. As a consequence, we provide correct asymptotic estimates on an old problem in convex geometry. Namely, we show that the simplex $S_{min}(K)$ of minimal volume enclosing a given convex body $K \subset \mathbb R^n$, fulfills the following inequality $$\left(\frac{vol(S_{min}(K))}{vol(K)}\right)^{1/n} \leq d \sqrt{n},$$ for some absolute constant $d>0$. Up to the constant, the estimate cannot be lessened.

math.MG↗

arXiv2019

Monomial convergence on $\ell_r$

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 $m_{Ψ_r}$ where the symbol $Ψ_r$ is given by $Ψ_r(n) := \log(n + 1)^{1 - \frac{1}{r}}$ for $n \in \mathbb N_0$. For the space of $m$-homogeneous polynomials on $\ell_r$, we prove that the set of monomial convergence $\mbox{mon} \mathcal P (^m \ell_r)$ contains the sequence space $\ell_{q}$ where $q=(mr')'$. Moreover, we show that for any $q\leq s<\infty$, the Lorentz sequence space $\ell_{q,s}$ lies in $\mbox{mon} \mathcal P (^m \ell_r)$, provided that $m$ is large enough. We apply our results to make an advance in the description of the set of monomial convergence of $H_{\infty}(B_{\ell_r})$ (the space of bounded holomorphic on the unit ball of $\ell_r$). As a byproduct we close the gap on certain estimates related with the \emph{mixed} unconditionality constant for spaces of polynomials over classical sequence spaces.

math.CV↗

arXiv2018

The sup-norm vs. the norm of the coefficients: equivalence constants for homogeneous polynomials

Let $A_{p,r}^m(n)$ be the best constant that fulfills the following inequality: for every $m$-homogeneous polynomial $P(z) = \sum_{|α|=m} a_α z^α$ in $n$ complex variables, $$\big( \sum_{|α|=m} |a_α|^{r} \big)^{1/r} \leq A_{p,r}^m(n) \sup_{z \in B_{\ell_p^n}} \big|P(z) \big| .$$ For every degree $m$, and a wide range of values of $ p,r \in [1,\infty]$ (including any $r$ in the case $p \in [1,2]$, and any $r$ and $p$ for the 2-homogeneous case), we give the correct asymptotic behavior of these constants as $n$ (the number of variables) tends to infinity. Remarkably, in many cases, extremal polynomials for these inequalities are not (as traditionally expected) found using classical random unimodular polynomials, and special combinatorial configurations of monomials are needed. Namely, we show that Steiner polynomials (i.e., $m$-homogeneous polynomials such that the multi-indices corresponding to the nonzero coefficients form partial Steiner systems), do the work for certain range of values of $p,r$. As a byproduct, we present some applications of these estimates to the interpolation of tensor products of Banach spaces, to the study of (mixed) unconditionality in spaces of polynomials and to the multivariable von Neumann's inequality.

math.FA↗

arXiv2018

Operator $p$-compact mappings

We introduce the class of operator $p$-compact mappings and completely right $p$-nuclear operators, which are natural extensions to the operator space framework of their corresponding Banach operator ideals. We relate these two classes, define natural operator space structures and study several properties of these ideals. We show that the class of operator $\infty$-compact mappings in fact coincides with a notion already introduced by Webster in the nineties (in a very different language). This allows us to provide an operator space structure to Webster's class.

math.FA↗

arXiv2017

Mixed Bohr radius in several variables

Let $K(B_{\ell_p^n},B_{\ell_q^n}) $ be the $n$-dimensional $(p,q)$-Bohr radius for holomorphic functions on $\mathbb C^n$. That is, $K(B_{\ell_p^n},B_{\ell_q^n}) $ denotes the greatest constant $r\geq 0$ such that for every entire function $f(z)=\sum_α c_α z^α$ in $n$-complex variables, we have the following (mixed) Bohr-type inequality $$\sup_{z \in r \cdot B_{\ell_q^n}} \sum_α | c_α z^α | \leq \sup_{z \in B_{\ell_p^n}} | f(z) |,$$ where $B_{\ell_r^n}$ denotes the closed unit ball of the $n$-dimensional sequence space $\ell_r^n$. For every $1 \leq p, q \leq \infty$, we exhibit the exact asymptotic growth of the $(p,q)$-Bohr radius as $n$ (the number of variables) goes to infinity.

math.CV↗

arXiv2017

Cluster values for algebras of analytic functions

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 Fréchet algebra $H_b(X)$ of holomorphic functions of bounded type on $X$ (more generally, for $H_b(U)$, the algebra of holomorphic functions of bounded type on a given balanced open subset $U \subset X$). We show that Cluster Value Theorems hold for all of these algebras whenever the dual of $X$ has the bounded approximation property. These results are an important advance in this problem, since the validity of these theorems was known only for trivial cases (where the spectrum is formed only by evaluation functionals) and for the infinite dimensional Hilbert space. As a consequence , we obtain weak analytic Nullstellensatz theorems and several structural results for the spectrum of these algebras.

math.FA↗

arXiv2015

Asymptotic estimates on the von Neumann inequality for homogeneous polynomials

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 \[ \|p(T_1, \dots, T_n)\|_{\mathcal L(\mathcal H)} \leq C_{k,q}(n) \; \sup\{ |p(z_1, \dots, z_n)| : \textstyle \sum_{i=1}^{n} \vert z_{i} \vert^{q} \leq 1 \}\,. \] For fixed $k$ and $q$, we study the asymptotic growth of the smallest constant $C_{k,q}(n)$ as $n$ (the number of variables/operators) tends to infinity. For $q = \infty$, we obtain the correct asymptotic behavior of this constant (answering a question posed by Dixon in the seventies). For $2 \leq q < \infty$ we improve some lower bounds given by Mantero and Tonge, and prove the asymptotic behavior up to a logarithmic factor. To achieve this we provide estimates of the norm of homogeneous unimodular Steiner polynomials, i.e. polynomials such that the multi-indices corresponding to the nonzero coefficients form partial Steiner systems.

math.FA↗

arXiv2015

$L^q$ dimensions and projections of random measures

We prove preservation of $L^q$ dimensions (for $1<q\le 2$) under all orthogonal projections for a class of random measures on the plane, which includes (deterministic) homogeneous self-similar measures and a well-known family of measures supported on $1$-variable fractals as special cases. We prove a similar result for certain convolutions, extending a result of Nazarov, Peres and Shmerkin. Recently many related results have been obtained for Hausdorff dimension, but much less is known for $L^q$ dimensions.

math.DS↗

arXiv2014

Energy integrals and metric embedding theory

For some centrally symmetric convex bodies $K\subset \mathbb R^n$, we study the energy integral $$ \sup \int_{K} \int_{K} \|x - y\|_r^{p}\, dμ(x) dμ(y), $$ where the supremum runs over all finite signed Borel measures $μ$ on $K$ of total mass one. In the case where $K = B_q^n$, the unit ball of $\ell_q^n$ (for $1 < q \leq 2$) or an ellipsoid, we obtain the exact value or the correct asymptotical behavior of the supremum of these integrals. We apply these results to a classical embedding problem in metric geometry. We consider in $\mathbb R^n$ the Euclidean distance $d_2$. For $0 < α< 1$, we estimate the minimum $R$ for which the snowflaked metric space $(K, d_2^α)$ may be isometrically embedded on the surface of a Hilbert sphere of radius $R$.

math.MG↗

arXiv2014

Coincidence of extendible vector-valued ideals with their minimal kernel

We provide coincidence results for vector-valued ideals of multilinear operators. More precisely, if $\mathfrak A$ is an ideal of $n$-linear mappings we give conditions for which the following equality $\mathfrak A(E_1,\dots,E_n;F) = {\mathfrak A}^{min}(E_1,\dots,E_n;F)$ holds isometrically. As an application, we obtain in many cases that the monomials form a Schauder basis on the space $\mathfrak A(E_1,\dots,E_n;F)$. Several structural and geometric properties are also derived using this equality. We apply our results to the particular case where $\mathfrak A$ is the classical ideal of extendible or Pietsch-integral multilinear operators. Similar statements are given for ideals of vector-valued homogeneous polynomials.

math.FA↗