arXiv subjects
Publications and source records attributed to Daniel Seco.
We consider shift operators on infinite directed trees. We prove that under very mild assumptions on the structure of the tree, the resulting operator is universal (in the sense of Rota). We show connections between the spectrum of the operator and the frequency of branching in the tree.
Lefèvre, Li, Queffélec, and Rodríguez-Piazza [12] recently characterized the sequences $β$ for which the weighted Hardy space $H^2(β)$ has the same bounded composition operators as the Hardy space $H^2$. We investigate the analogous questions for the compactness and the $p$-Schatten classes: for which sequences $β$ does the space $H^2(β)$ have the same compact or $p$-Schatten class composition operators as $H^2$?
We study optimal polynomial approximants in spaces of Dirichlet-type over the bidisc. Given a nonzero function f, the linear optimal polynomial approximant (opa) to its reciprocal is the affine polynomial p for which pf is closest to the constant function 1 in the space norm. We prove that, for every positive value of the Dirichlet-type parameter, there exists a symmetric polynomial f with no zeros on the closed bidisc, while the corresponding linear optimal approximant has a zero inside the bidisc. This provides polynomial counterexamples to the version of the Weak Shanks Conjecture on these spaces. The construction also extends to higher-dimensional polydiscs and to anisotropic Dirichlet-type spaces.
Let $H$ be a reproducing kernel Hilbert space over the unit disk $\mathbb{D}$, where analytic monomials span a dense subset. Given $\mathcal{N} \subseteq\mathbb{Z}_+$ and $Λ\subseteq \mathbb{D}$ we say that $(Λ,\mathcal{N})$ is a uniqueness pair for $H$ if $Λ$ is a uniqueness set for the subspace of $H$ spanned by $\{z^n:\;n\in\mathcal{N}\}$. We examine uniqueness pairs in the Dirichlet-type spaces $\mathbb{D}_α$, $0\leqα\leq1$. We prove two complementary results. First, if $\mathcal{N}$ contains sufficiently long finite arithmetic progressions with fixed gap size, then no sequence $Λ$ tending sufficiently rapidly to the boundary forms a uniqueness pair with $\mathcal{N}$. Second, if $\mathcal{N}$ satisfies a suitable arithmetic sparsity condition then one can construct uniqueness pairs $(Λ,\mathcal{N})$ with the points of $Λ$ tending to the boundary arbitrarily fast.
Given a function $b$, holomorphic on the disc and bounded by 1, one can construct an associated reproducing kernel Hilbert space called the de Branges--Rovnyak space $H(b)$. We explore representations of such spaces via descriptions of the corresponding families of orthogonal polynomials. We find relevant structures in the linear systems involved in a diversity of cases when $b$ is rational. We also establish a form of invariance under some composition operators on $H(b)$ spaces.
We show the existence of singular inner functions that are cyclic in some Besov-type spaces of analytic functions over the unit disc. Our sufficient condition is stated only in terms of the modulus of smoothness of the underlying measure. Such singular inner functions are cyclic also in the space $\ell^p_A$ of holomorphic functions with coefficients in $\ell^p$. This can only happen for measures that place no mass on any Beurling-Carleson set.
We explore the connection between two seemingly distant fields: the set of cyclic functions $f$ in a Hilbert space of analytic functions over the unit disc $\D$, on the one hand, and the families of orthogonal polynomials for a weight on the unit circle $\T$ (OPUC), on the other. This link is established by so-called Optimal Polynomial Approximants (OPA) to $1/f$, that is, polynomials $p_n$ minimizing the norm of $1-p_nf$, among all polynomials $p_n$ of degree up to a given $n$. Here, we focus on the particular case of the Hardy space, and an electrostatic interpretation of the zeros of those OPA (and thus, of the corresponding OPUC) is studied. We find the electrostatic laws explaining the position of such zeros for a reduced but significant class of examples. This represents the first step towards a research plan proposed over a decade ago to understand zeros of OPA through their potential theoretic properties.
We disprove a conjecture by Ye and Lim, by showing that there are $3 \times 3$ complex matrices which can't be expressed as the product of two Toeplitz matrices of the same size. We also improve previous estimates by Ye and Lim on the minimum number of Toeplitz matrices needed to factor any $n \times n$ matrix, for low values of $n$.
We characterize bounded multiplication operators in weighted Dirichlet spaces that are power bounded, Cesàro bounded and uniformly Kreiss. Moreover, we show the equivalence in such spaces between mean ergodicity and Cesàro boundedness for multiplication operators. We perform the same study for adjoints of multiplication operators. As a particular example, we obtain a uniform mean ergodic multiplication operator in Dirichlet spaces that fails to be power bounded.
We build on a characterization of inner functions $f$ due to Le, in terms of the spectral properties of the operator $V=M_f^*M_f$ and study to what extent the cyclicity on weighted Hardy spaces $H^2_ω$ of the function $z \mapsto a-z$ can be inferred from the spectral properties of analogous operators $V_a$. We describe several properties of the spectra that hold in a large class of spaces and then, we focus on the particular case of Bergman-type spaces, for which we describe completely the spectrum of such operators and find all eigenfunctions.
To every non-extreme point $b$ of the unit ball of $\hil^\infty$ of the unit disk there corresponds a Pythagorean mate, a bounded outer function $a$ satisfying the equation $|a|^2 + |b|^2 = 1$ on the boundary of the disk. We study universal, i.e., simultaneous multipliers for families of de Branges-Rovnyak spaces $\hb$, and develop a general framework for this purpose. Our main results include a new proof of the Davis-McCarthy universal multiplier theorem for the class of all non-extreme spaces $\hb$, a characterization of the Lipschitz classes as the universal multipliers for spaces $\hb$ for which the quotient $b/a$ is contained in a Hardy space, and a similar characterization of the Gevrey classes as the universal multipliers for spaces $\hb$ for which $b/a$ is contained in a Privalov class.
We study zero-free regions of the Riemann zeta function $ζ$ related to an approximation problem in the weighted Dirichlet space $D_{-2}$ which is known to be equivalent to the Riemann Hypothesis since the work of Báez-Duarte. We prove, indeed, that analogous approximation problems for the standard weighted Dirichlet spaces $D_α$ when $α\in (-3,-2)$ give conditions so that the half-plane $\{s \in \mathbb{C}: \Re (s) > -\frac{α+1}{2}\}$ is also zero-free for $ζ$. Moreover, we extend such results to a large family of weighted spaces of analytic functions $\ell^p_α$. As a particular instance, in the limit case $p=1$ and $α=-2$, we provide a new equivalent formulation of the Prime Number Theorem.
We give an example of a function $f$ non-vanishing in the closed bidisk and the affine polynomial minimizing the norm of $1-pf$ in the Hardy space of the bidisk among all affine polynomials $p$. We show that this polynomial vanishes inside the bidisk. This provides a counterexample to the weakest form of a conjecture due to Shanks that has been open since 1980, with applications that arose from digital filter design. This counterexample has a simple form and follows naturally from [7], where the phenomenon of zeros seeping into the unit disk was already observed for similar minimization problems in one variable.
We study conditions for containment of a given space $X$ of analytic functions on the unit disk $\mathbb{D}$ in the de Branges-Rovnyak space $\mathcal{H}(b)$. We deal with the non-extreme case in which $b$ admits a Pythagorean mate $a$, and derive a multiplier boundedness criterion on the function $ϕ= b/a$ which implies the containment $X \subset \mathcal{H}(b)$. With our criterion, we are able to characterize the containment of the Hardy space $\mathcal{H}^p$ inside $\mathcal{H}(b)$, for $p \in [2, \infty]$. The end-point cases have previously been considered by Sarason, and we show that in his result, stating that $ϕ\in \mathcal{H}^2$ is equivalent to $\mathcal{H}^\infty \subset \mathcal{H}(b)$, one can in fact replace $\mathcal{H}^\infty$ by BMOA. We establish various other containment results, and study in particular the case of the Dirichlet space $\mathcal{D}$, containment of which is characterized by a Carleson measure condition. In this context, we show that matters are not as simple as in the case of the Hardy spaces, and we carefully work out an example.
We prove the property that a function is cyclic (resp., non-cyclic) is not preserved by norm convergence in Dirichlet-type spaces $D_α$, and show how other significant quantities for cyclicity do remain preserved under the limit of convergent sequences in $D_α$, providing a quantitative view of this convergence issue.
In the study of the cyclicity of a function $f$ in reproducing kernel Hilbert spaces an important role is played by sequences of polynomials $\{p_n\}_{n\in \mathbb{N}}$ called \emph{optimal polynomial approximants} (o.p.a.). For many such spaces and when the functions $f$ generating those o.p.a. are polynomials without zeros inside the disk but with some zeros on its boundary, we find that the weakly asympotic distribution of the zeros of $1-p_nf$ is the uniform measure on the unit circle.
The study of inner and cyclic functions in $\ell^p_A$ spaces requires a better understanding of the zeros of the so-called optimal polynomial approximants. We determine that a point of the complex plane is the zero of an optimal polynomial approximant for some element of $\ell^p_A$ if and only if it lies outside of a closed disk (centered at the origin) of a particular radius which depends on the value of $p$. We find the value of this radius for $p\neq 2$. In addition, for each positive integer $d$ there is a polynomial $f_d$ of degree at most $d$ that minimizes the modulus of the root of its optimal linear polynomial approximant. We develop a method for finding these extremal functions $f_d$ and discuss their properties. The method involves the Lagrange multiplier method and a resulting dynamical system.
In previous works, an approach to the study of cyclic functions in reproducing kernel Hilbert spaces has been presented, based on the study of so called \emph{optimal polynomial approximants}. In the present article, we extend such approach to the (non-Hilbert) case of spaces of analytic functions whose Taylor coefficients are in $\ell^p(ω)$, for some weight $ω$. When $ω=\{(k+1)^α\}_{k\in \mathbb{N}}$, for a fixed $α\in \mathbb{R}$, we derive a characterization of the cyclicity of polynomial functions and, when $1<p<\infty$, we obtain sharp rates of convergence of the optimal norms.