Réunion d'hiver SMC 2024
Vancouver/Richmond, 29 novembre - 2 decembre 2024
Operator Theory, Function Theory, and Geometry: Connections to Corona Problems and Geometric Analysis
Org: Alexander Brudnyi et Mahishanka Withanachchi (University of Calgary)
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Org: Alexander Brudnyi et Mahishanka Withanachchi (University of Calgary)
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- AKRAM ALDROUBI, Vanderbilt University
Dynamical sampling: source term recovery and frames [PDF]
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In this talk, I will address the problem of recovering a source terms in a discrete dynamical system represented by $x_{n+1} = Ax_n + w$, where $x_n$ is the $n$-th state in a Hilbert space $\mathcal{H}$, $A$ is a bounded linear operator in $\mathcal{B}(\mathcal{H})$, and $w$ is a source term within a closed subspace $W$ of $\mathcal H$. The focus is on the stable recovery of $w$ using time-space sample measurements formed by inner products with vectors from a Bessel system $\mathcal{G} \subset \mathcal{H}$. These types of results may be relevant to applications such as environmental monitoring, where precise source identification is critical. This work is in collaboration with Rocio Diaz Martin Le Gong, Javad Mashreghi, and Ivan Medri.
- ILIA BINDER, University of Toronto
Conformal Dimension of Planar fractals. [PDF]
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The \emph{conformal dimension} of a set is the minimal Hausdorff dimension of its quasisymmetric image. In this talk, I will discuss the conformal dimensions of various planar fractals, including Bedford-McMullen sets and self-affine fractal percolation clusters. I will also demonstrate that the Brownian graph is \emph{minimal}, meaning its conformal dimension is $3/2$, which is also its Hausdorff dimension.
This work is a collaboration with Hrant Hakobyan from Kansas State University and Wenbo Li from Peking University.
- LUDOVICK BOUTHAT, Université Laval
Exploring Hadamard multipliers on weighted Dirichlet spaces through $L$-matrices [PDF]
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The Hadamard product of two power series is obtained by multiplying them coefficientwise. In 2020, Mashreghi and Ransford characterized those power series that act as Hadamard multipliers on all weighted Dirichlet spaces on the disk with superharmonic weight. These power series correspond to those whose associated $L$-matrix defines a bounded operator on $\ell^2$. An $L$-matrix is an infinite matrix $\mathcal{L}$ whose entries are of the form $\mathcal{L}_{i,j} = a_{\max\{i,j\}}$ for some complex sequence $(a_n)_{n\geq 0}$. In this talk, we present several conditions on the sequence $(a_n)_{n\geq0}$ for $\mathcal{L}$ to be a bounded operator on $\ell^2$ and we present a particular set of $L$-matrices for which we are able to exactly determine the norm.
This work is a collaboration with Javad Mashreghi.
- ALEX BRUDNYI, University of Calgary
Runge-Type Approximation Theorem for Banach-valued $ H^\infty$ Functions on a Polydisk [PDF]
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Let $\mathbb D^n\subset\mathbb C^n$ be the open unit polydisk, $K\subset\mathbb D^n$ be an $n$-ary Cartesian product of planar sets, and $\widehat U\subset \mathfrak M^n$ be an open neighbourhood of the closure $\bar K$ of $K$ in $\mathfrak M^n$, where $\mathfrak M$ is the maximal ideal space of the algebra $H^\infty$ of bounded holomorphic functions on $\mathbb D$. Let $X$ be a complex Banach space and $H^\infty(V,X)$ be the space of bounded $X$-valued holomorphic functions on an open set $V\subset\mathbb D^n$.
We show that any $f\in H^\infty(U,X)$, where $U=\widehat U\cap\mathbb D^n$, can be uniformly approximated on $K$ by ratios $h/b$, where $h\in H^\infty(\mathbb D^n,X)$ and $b$ is the product of interpolating Blaschke products such that $\inf_K |b|>0$. Moreover,
if $\bar K$ is contained in a compact holomorphically convex subset of $\widehat U$, then $h/b$ above can be replaced by $h$ for any $f$. The results follow from a new constructive Runge-type approximation theorem for Banach-valued holomorphic functions on open subsets of $\mathbb D$ and extend the fundamental results of Suarez on Runge-type approximation for analytic germs on compact subsets of $\mathfrak M$. They can also be applied to the long-standing corona problem which asks whether $\mathbb D^n$ is dense in the maximal ideal space of $H^\infty(\mathbb D^n)$ for all $n\ge 2$.
- RAPHAEL CLOUATRE, University of Manitoba
Joint spectra and annihilators in multivariate operator theory [PDF]
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For an appropriately regular, single Hilbert space contraction $T$, it is known that the spectrum can be described in terms of the annihilator, that is the ideal $\text{Ann}(T)$ of bounded holomorphic functions $f$ on the unit disc satisfying $f(T)=0$. Indeed, the spectrum coincides with the so-called support of $\text{Ann}(T)$. In this talk, we explore the extent to which a similar statement is valid for commuting tuples of operators $T=(T_1,T_2,\ldots,T_d)$. The corresponding multivariate notion of support for $\text{Ann}(T)$ is rather subtle. We will give a more concrete description of the support in terms of the zero set of $\text{Ann}(T)$ when it is assumed that the underlying space of holomorphic functions has the Corona property.
- DAMIR KINZEBULATOV, Université Laval
- PIERRE OLIVIER, UQTR
- MARIA PEREYRA, New Mexico
- ERIC SAWYER, McMaster University
Probabilistic and Deterministic Fourier Extension [PDF]
- PIERRE OLIVIER, UQTR
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We discuss the proof of the probabilistic Fourier extension theorem, and possible applications to the deterministic conjecture.
- KRYSTAL TAYLOR, Ohio State
- WILLIAM VERREAULT, University of Toronto
The Cesàro Operator on local Dirichlet spaces [PDF]
- WILLIAM VERREAULT, University of Toronto
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The family of Cesàro operators $\sigma_n^\alpha$, $n \geq 0$ and $0\leq \alpha \leq 1$, consists of finite rank operators on Banach spaces of
analytic functions on the open unit disc. We investigate these operators as
they act on the local Dirichlet spaces $\mathcal{D}_\zeta$. It is
well-established that they provide a linear approximation scheme when
$\alpha > \frac{1}{2}$, with the threshold value $\alpha = \frac{1}{2}$
being optimal. We strengthen this result by deriving precise asymptotic values
for the norm of these operators when $\alpha \leq \frac{1}{2}$, corresponding
to the breakdown of approximation schemes. Additionally, we establish upper
and lower estimates for the norm when $\alpha > \frac{1}{2}$.
This is joint work with Eugenio Dellepiane, Javad Mashreghi, and Mostafa Nasri.
- MAHISHANKA WITHANACHCHI, University of Calgary
Vanishing Cohomology and the Corona Problem for the Algebra of Bounded Holomorphic Functions on the Polydisk [PDF]
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In this talk, we study the Corona problem for the Banach algebra $H^\infty(\mathbb{D}^n)$ of bounded holomorphic functions on the polydisk $\mathbb{D}^n \subset \mathbb{C}^n$. In this setting, the Corona problem asks whether the polydisk $\mathbb{D}^n$ is dense in the Gelfand topology in the maximal ideal space of $H^\infty(\mathbb{D}^n)$. We present new necessary and sufficient conditions under which the problem can be solved. An important part of our work is a new result on the vanishing of the first cohomology of a sheaf of germs of holomorphic functions on the $n$-fold Cartesian product of the maximal ideal space of $H^\infty(\mathbb{D})$. Our method is based on a new important result on the solution of special $\overline{\partial}$ equations on a polydisk. This is a joint work with Alex Brudnyi.
- ZHICHUN ZHAI, MacEwan University
Stengthened Fractional Sobolev Inequalities and Geometric Inequalities [PDF]
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This study has two primary objectives. The first is to enhance fractional Sobolev-type inequalities in Besov spaces using the framework of classical Lorentz spaces. In this process, we establish that the Sobolev inequality in Besov spaces is equivalent to the fractional Hardy inequality and an iso-capacitary-type inequality.
The second objective is to strengthen fractional Sobolev-type inequalities in Besov spaces through capacitary Lorentz spaces associated with Besov capacities. To achieve this, we first analyze the embedding of the associated capacitary Lorentz space into the classical Lorentz space. Subsequently, we establish the embedding of the Besov space into the capacitary Lorentz space. Additionally, we demonstrate that these embeddings are intricately connected to iso-capacitary-type inequalities, interpreted through a newly introduced fractional $(\beta, p, q)-$perimeter. Furthermore, we provide characterizations of more general Sobolev-type inequalities in Besov spaces.
- NINA ZORBOSKA, University of Manitoba
Hankel measures and Hankel type operators on weighted Dirichlet spaces [PDF]
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I will talk about Hankel measures and the boundedness of measure induced Hankel type operators on weighted Dirichlet spaces, extending the known results for the cases of the classical Hardy and Dirichlet spaces. The approach relies on recent results on weak products of complete Nevanlinna-Pick reproducing kernel Hilbert spaces.