# Past Functional Analysis Seminar

## Further Information:

ABSTRACT: Given a metric measure space $(X,d,\mu)$, its doubling constant is given by

$$

C_\mu=\sup_{x\in X, r>0} \frac{\mu(B(x,2r))}{\mu(B(x,r))},

$$

where $B(x,r)$ denotes the open ball of radius $r$ centered at $x$. Clearly, $C_\mu\geq1$, and in the case $X$ reduces to a singleton $C_\mu=1$. One might think that for a metric space with more than one point, the constant $C_\mu$ could be very close to one. However, we will show that in general $C_\mu\geq2$. The talk is based on a joint work with Javier Soria (Barcelona).

## Further Information:

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.

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.

## Further Information:

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.

I will present the general strategy for classifying separable, nuclear, purely infinite C*-algebras

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).

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.