Past Functional Analysis Seminar

2 November 2018
11:00
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).

  • Functional Analysis Seminar
2 November 2018
09:30
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.

  • Functional Analysis Seminar
1 November 2018
17:00
David Seifert
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.

  • Functional Analysis Seminar
1 November 2018
14:30
to
17:00
Sylvie Monniaux
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”)

 

 

 

 

 

 

  • Functional Analysis Seminar
22 May 2018
17:00
Karl-Mikael Perfekt
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.

  • Functional Analysis Seminar
15 May 2018
17:00
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).
 

  • Functional Analysis Seminar
8 May 2018
17:00
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.

  • Functional Analysis Seminar

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