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Forthcoming events in this series
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Extensions of Uniform Algebras
The theory of algebraic extensions of commutative Banach algebras is well established and have been used to solve many problems. In his thesis, Cole constructed algebraic extensions of a certain uniform algebra to give a counterexample to the peak point conjecture. Cole’s method for extending uniform algebras ensures that certain properties of the original algebra are preserved by the extension. In this talk, we discuss the general theory of uniform algebra extensions and a certain class of uniform algebra extensions which generalise Cole’s construction.
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The behaviour of the $(p, q)$-version of Fourier's series
Abstract
Consider a periodic function $f$, such that its restriction to the unit segment lies in the Banach space $L^2=L^2(0,1)$. Denote by $S$ the family of dilations $f(nx)$ for all $n$ positive integer. The purpose of this talk is to discuss the following question: When does $S$ form a Riesz basis of $L^2$?
In this talk, we will present a new \textit{mutli-term} criteria for determining Riesz basis properties of $S$ in $L^2$. This method was established in [L. Boulton, H. Melkonian; arXiv: 1708.08545 J. (2017), to appear at the Journal of Analysis and its Applications (ZAA)] and it relies on a general framework developed by Hedenmalm, Lindqvist and Seip about 20 years ago, which turns the basis question into one about the localisation of the zeros and poles of a corresponding analytic multiplier. Our results improve upon various criteria formulated previously, which give sufficient conditions for invertibility of the multiplier in terms of sharp estimates on the Fourier coefficients.
17:00
Admissibility problem of some classes of state-delayed systems
Beginning with a short introduction and a review of Hilbert space
techniques used in the admissibility analysis of dynamical systems,
we will focus on state-delayed systems.
Using the "lifting" method to reformulate the problem, we will firstly
analyse a retarded delay system assuming only contraction property
of the undelayed semigroup. Next, we will turn our attention to problems
where more can be said about the underlying semigroup.
In particular, we will investigate diagonal systems.
This talk will present results of a joint work with Jonathan Partington.
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Classification of purely infinite C*-algebras
Abstract
I will present the general strategy for classifying separable, nuclear, purely infinite C*-algebras
Fourier multipliers and stability of semigroups
Abstract
This is part of a meeting of the North British Functional Analysis Seminar
In this talk I will present some new $L_p$-$L_q$-Fourier multiplier theorems which hold for operator-valued symbols under geometric restrictions on the underlying Banach spaces such as (Fourier) (co)type. I will show how the multiplier theorems can be applied to obtain new stability results for semigroups arising in evolution equations. This is based on joint work with Jan Rozendaal (ANU, Canberra).
Square functions and random sums and their role in the analysis of Banach spaces
Abstract
This is part of a meeting of the North British Functional Analysis Seminar.
In this talk I will present an overview on generalized square functions in Banach spaces and some of their recent uses in “Analysis in Banach Spaces”. I will introduce the notions of $R$-boundedness and $\gamma$-radonifying operators and discuss their origins and some of their applications to harmonic analysis, functional calculus, control theory, and stochastic analysis.
Optimal rates of decay for semigroups on Hilbert spaces
Abstract
This talk is associated with the NBFAS meeting.
We discuss the quantitative asymptotic behaviour of operator semigroups. Batty and Duyckaerts obtained upper and lower bounds on the rate of decay of a semigroup given bounds on the resolvent growth of the semigroup generator. They conjectured that in the Hilbert space setting and for the special case of polynomial resolvent growth it is possible to improve the upper bound so as to yield the exact rate of decay up to constants. This conjecture was proved to be correct by Borichev and Tomilov, and the conclusion was extended by Batty, Chill and Tomilov to certain cases in which the resolvent growth is not exactly polynomial but almost. In this talk we extend their result by showing that one can improve the upper bound under a significantly milder assumption on the resolvent growth. This result is optimal in a certain sense. We also discuss how this improved result can be used to obtain sharper estimates on the rate of energy decay for a wave equation subject to viscoelastic damping at the boundary. The talk is based on joint work with J. Rozendaal and R. Stahn.
Potential operators, analyticity and bounded holomorphic functional calculus for the Stokes operator
Abstract
This is part of a meeting of the North British Functional Aanlysis Seminar. There will be a tea break (15:30-16:00)
In a first talk, I shall recall the basic definitions and properties of ${\mathcal{H}}^\infty}$ functional calculus. I shall show how a second order problem can be reduced to a first order system and how to construct potential operators.
In a second talk, we will see how to use potential operators for the specific problem of the Stokes operator with the so-called “natural” boundary conditions in non smooth domains.
Most parts which will be presented are taken from a joint work with Alan McIntosh (to be published soon in the journal "Revista Matematica Iberoamericana”)
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On the decay rate for the wave equation with viscoelastic boundary damping
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On the spectral resolution of the Neumann-Poincare operator
Abstract
The Neumann-Poincare (NP) operator (or the double layer potential) has classically been used as a tool to solve the Dirichlet and Neumann problems of a domain. It also serves as a prominent example in non-self adjoint spectral theory, due to its unexpected behaviour for domains with singularities. Recently, questions from materials science have revived interest in the spectral properties of the NP operator on domains with rough features. I aim to give an overview of recent developments, with particular focus on the NP operator's action on the energy space of the domain. The energy space framework ties together Poincare’s efforts to solve the Dirichlet problem with the operator-theoretic symmetrisation theory of Krein. I will also indicate recent work for domains in 3D with conical points. In this situation, we have been able to describe the spectrum both for boundary data in $L^2$ and for data in the energy space. In the former case, the essential spectrum consists of the union of countably many self-intersecting curves in the plane, and outside of this set the index may be computed as the winding number with respect to the essential spectrum. In the latter case the essential spectrum consists of a real interval.
17:00
Why do circles in the spectrum matter?
Abstract
I plan to present several results linking the numerical range of a Hilbert space operator to the circle structure of its spectrum. I'll try to explain how the numerical ranges approach helps to unify, extend or supplement several results where the circular structure of the spectrum is crucial, e.g. Arveson's theorem on almost-wandering vectors of unitary actions and Hamdan's recent result on supports of Rajchman measures. Moreover, I'll give several applications of the approach to new operator-theoretical constructions inverse in a sense to classical power dilations. If time permits, I'll also address the same or similar issues in a more general setting of operator tuples. This is joint work with V. M\" uller (Prague).
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A quantified Tauberian theorem for the Laplace-Stieltjes transform
Abstract
We consider a vector-valued function $f: \mathbb{R}_+ \to X$ which is locally of bounded variation and give a decay rate for $|A(t)|$ for increasing $t$ under certain conditions on the Laplace-Stieltjes transform $\widehat{dA}$ of $A$. For this, we use a Tauberian condition inspired by the work of Ingham and Karamata and a contour integration method invented by Newman. Our result is a generalisation of already known Tauberian theorems for bounded functions and is applicable to Dirichlet series. We will say something about the connection between the obtained decay rates and number theory.
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Vectorial Hankel operators, Carleson embeddings, and notions of BMOA
Abstract
Let $\mathrm{BMOA}_{\mathcal{NP}}$ denote the space of operator-valued analytic functions $\phi$ for which the Hankel operator $\Gamma_\phi$ is $H^2(\mathcal{H})$-bounded. Obtaining concrete characterizations of $\mathrm{BMOA}_{\mathcal{NP}}$ has proven to be notoriously hard. Let $D^\alpha$ denote differentiation of fractional order $\alpha$. Motivated originally by control theory, we characterize $H^2(\mathcal{H})$-boundedness of $D^\alpha\Gamma_\phi$, where $\alpha>0$, in terms of a natural anti-analytic Carleson embedding condition. We obtain three notable corollaries: The first is that $\mathrm{BMOA}_{\mathcal{NP}}$ is not characterized by said embedding condition. The second is that when we add an adjoint embedding condition, we obtain a sufficient but not necessary condition for boundedness of $\Gamma_\phi$ . The third is that there exists a bounded analytic function for which the associated anti-analytic Carleson embedding is unbounded. As a consequence, boundedness of an analytic Carleson embedding does not imply that the anti-analytic ditto is bounded. This answers a question by Nazarov, Pisier, Treil, and Volberg.
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Group C*-algebras and some examples
Abstract
Given a locally compact group G, the group C*-algebra is defined by taking the completion of $L^1(G)$ with respect to the C*-norm given by the irreducible unitary representations of G. However, if the group is not abelian, there is no known concrete description of its group C*-algebra. In my talk, I will briefly introduce the group C*-algebras and then give some examples arisen from solvable Lie groups
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Transference methods for spectral properties of Helson operators
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Extension of suboperators and the generalized Schur complement
Abstract
Our long term plan is to develop a unified approach to prove decomposition theorems in different structures. In our anti-dual pair setting, it would be useful to have a tool which is analogous to the so-called Schur complementation. To this aim, I will present a suitable generalization of the classical known Krein - von Neumann extension.
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Remainders in the Ingham-Karamata Tauberian theorem
Abstract
The classical Ingham-Karamata Tauberian theorem has many applications in different fields of mathematics, varying from number theory to $C_0$-semigroup theory and is considered to be one of the most important Tauberian theorems. We will discuss how to obtain remainder estimates in the theorem if one strengthens the assumptions on the Laplace transform. Moreover, we will give new (remainder) versions of this theorem under the more general one-sided Tauberian condition of $\rho(x) \ge −f(x)$ where $f$ is an arbitrary function satisfying some regularity assumptions. The talk is based on collaborative work with Jasson Vindas.
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Facial structure of the unit ball in a JB*-triple
Abstract
We present recent results on the connections existing between the facial
structure of the unit ball in a JB*-triple and the lattice of tripotents in its
bidual.
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Harmonic measure, absolute continuity, and rectifiability
Abstract
For reasonable domains $\Omega\subseteq\mathbb{R}^{d+
1}$, and given some boundary data $f\in C(\partial\Omega)$, we can solve the Dirichlet problem and find a harmonic function $u_{f}$ that agrees with $f$ on $\partial\Omega$. For $x_{0}\in \Omega$, the association $f\rightarrow u_{f}(x_{0})$. is a linear functional, so the Riesz Representation gives us a measure $\omega_{\Omega}^{x_{0}}$ on $\partial\Omega$ called the harmonic measure with pole at $x_{0}$. One can also think of the harmonic measure of a set $E\subseteq \partial\Omega$ as the probability that a Brownian motion of starting at $x_{0}$ will first hit the boundary in $E$. In this talk, we will survey some very recent results about the relationship between the measure theoretic behavior of harmonic measure and the geometry of the boundary of its domain. In particular, we will study how absolute continuity of harmonic measure with respect to $d$-dimensional Hausdorff measure implies rectifiability of the boundary and vice versa.
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Discrete Fourier Analysis and spectral properties
Abstract
We present some recent results on the study of Schatten-von Neumann properties for
operators on compact manifolds. We will explain the point of view of kernels and full symbols. In both cases
one relies on a suitable Discrete Fourier analysis depending on the domain.
We will also discuss about operators on $L^p$ spaces by using the notion of nuclear operator in the sense of
Grothendieck and deduce Grothendieck-Lidskii trace formulas in terms of the matrix-symbol. We present examples
for fractional powers of differential operators. (Joint work with Michael Ruzhansky)
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Asymptotics for infinite systems of differential equations
Abstract
In this presentation we study the asymptotic behaviour of infinite systems of coupled linear ordinary differential equations. Each subsystem has identical dynamics that are only dependent on the states of its immediate neigbours. Examples of such systems in particular include the infinite "robot rendezvous problem" and the "platoon system" that are used to approximate the dynamics of large configurations of vehicles. In the presentation introduce novel methods for studying the spectral properties and stability of infinite systems of differential equations. The latter question is particularly interesting due to the fact that the systems in our class are known to lack uniform exponential stability. As our main results, we introduce general conditions for strong stability and derive rational rates of convergence for the solutions using recent results in the theory of nonuniform stability of strongly continuous semigroups.