arXiv subjects

Santiago Muro

Publications and source records attributed to Santiago Muro.

31 records · Page 2Linked to original sources

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

Orbits of homogeneous polynomials on Banach spaces

We study the dynamics induced by homogeneous polynomials on Banach spaces. It is known that no homogeneous polynomial defined on a Banach space can have a dense orbit. We show, a simple and natural example of a homogeneous polynomial with an orbit that is at the same time $d$-dense (the orbit meets every ball of radius $d$), weakly dense and such that $Γ\cdot Orb_P(x)$ is dense for every $Γ\subset \mathbb C$ that is either unbounded or that has 0 as an accumulation point. Moreover we generalize the construction to arbitrary infinite dimensional separable Fréchet spaces. To prove this we study Julia sets of homogeneous polynomials on Banach spaces.

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

Hypercyclic homogeneous polynomials on $H(\mathbb C)$

It is known that homogeneous polynomials on Banach spaces cannot be hypercyclic, but there are examples of hypercyclic homogeneous polynomials on some non-normable Fréchet spaces. We show the existence of hypercyclic polynomials on $H(\mathbb C)$, by exhibiting a concrete polynomial which is also the first example of a frequently hypercyclic homogeneous polynomial on any $F$-space. We prove that the homogeneous polynomial on $ H(\mathbb C)$ defined as the product of a translation operator and the evaluation at 0 is mixing, frequently hypercyclic and chaotic. We prove, in contrast, that some natural related polynomials fail to be hypercyclic.

math.FA↗

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

Hypercyclic behavior of some non-convolution operators on $H(\mathbb{C}^N)$

We study hypercyclicity properties of a family of non-convolution operators defined on spaces of holomorphic functions on $\mathbb{C}^N$. These operators are a composition of a differentiation operator and an affine composition operator, and are analogues of operators studied by Aron and Markose on $H(\mathbb{C})$. The hypercyclic behavior is more involved than in the one dimensional case, and depends on several parameters involved.

math.FA↗

arXiv2015

An integral formula for multiple summing norms of operators

We prove that the multiple summing norm of multilinear operators defined on some $n$-dimensional real or complex vector spaces with the $p$-norm may be written as an integral with respect to stables measures. As an application we show inclusion and coincidence results for multiple summing mappings. We also present some contraction properties and compute or estimate the limit orders of this class of operators.

math.FA↗

arXiv2014

Strongly mixing convolution operators on Fréchet spaces of holomorphic functions

A theorem of Godefroy and Shapiro states that non-trivial convolution operators on the space of entire functions on $\mathbb{C}^n$ are hypercyclic. Moreover, it was shown by Bonilla and Grosse-Erdmann that they have frequently hypercyclic functions of exponential growth. On the other hand, in the infinite dimensional setting, the Godefroy-Shapiro theorem has been extended to several spaces of entire functions defined on Banach spaces. We prove that on all these spaces, non-trivial convolution operators are strongly mixing with respect to a gaussian probability measure of full support. For the proof we combine the results previously mentioned and we use techniques recently developed by Bayart and Matheron. We also obtain the existence of frequently hypercyclic entire functions of exponential growth.

math.FA↗

arXiv2011

On algebras of holomorphic functions of a given type

We show that several spaces of holomorphic functions on a Riemann domain over a Banach space, including the nuclear and Hilbert-Schmidt bounded type, are locally $m$-convex Fréchet algebras. We prove that the spectrum of these algebras has a natural analytic structure, which we use to characterize the envelope of holomorphy. We also show a Cartan-Thullen type theorem.

math.FA↗

arXiv2011

Envelopes of holomorphy and extension of functions of bounded type

We study the extension of holomorphic functions of bounded type defined on an open subset of a Banach space, to larger domains. For this, we first characterize the envelope of holomorphy of a Riemann domain over a Banach space, with respect to the algebra of bounded type holomorphic functions, in terms of the spectrum of the algebra. We then give a simple description of the envelopes of balanced open sets and relate the concepts of domain of holomorphy and polynomial convexity. We show that for bounded balanced sets, extensions to the envelope are always of bounded type, and that this does not necessarily hold for unbounded sets, answering a question posed by Hirschowitz in 1972. We also consider extensions to open subsets of the bidual, present some Banach-Stone type results and show some properties of the spectrum when the domain is the unit ball of $\ell_p$.

math.FA↗

arXiv2010

Every Banach ideal of polynomials is compatible with an operator ideal

We show that for each Banach ideal of homogeneous polynomials, there exists a (necessarily unique) Banach operator ideal compatible with it. Analogously, we prove that any ideal of $n$-homogeneous polynomials belongs to a coherent sequence of ideals of $k$-homogeneous polynomials.

math.FA↗

arXiv2010

Holomorphic Functions and polynomial ideals on Banach spaces

Given $\u$ a multiplicative sequence of polynomial ideals, we consider the associated algebra of holomorphic functions of bounded type, $H_{b\u}(E)$. We prove that, under very natural conditions verified by many usual classes of polynomials, the spectrum $M_{b\u}(E)$ of this algebra "behaves" like the classical case of $M_{b}(E)$ (the spectrum of $H_b(E)$, the algebra of bounded type holomorphic functions). More precisely, we prove that $M_{b\u}(E)$ can be endowed with a structure of Riemann domain over $E"$ and that the extension of each $f\in H_{b\u}(E)$ to the spectrum is an $\u$-holomorphic function of bounded type in each connected component. We also prove a Banach-Stone type theorem for these algebras.

math.FA↗