Past Functional Analysis Seminar

23 April 2013
17:00
to
18:08
David Edwards
Abstract
We consider a vector lattice L of bounded real continuous functions on a topological space X that separates the points of X and contains the constant functions. A notion of tightness for linear functionals is defined, and by an elementary argument we prove with the aid of the classical Riesz representation theorem that every tight continuous linear functional on L can be represented by integration with respect to a Radon measure. This result leads incidentally to an simple proof of Prokhorov’s existence theorem for the limit of a projective system of Radon measures.
  • Functional Analysis Seminar
5 March 2013
17:00
to
18:16
Olivia Constantin
Abstract
We consider spaces of entire functions that are $p$-integrable with respect to a radial weight. Such spaces are usually called Fock spaces, and a classical example is provided by the Gaussian weight. It turns out that a function belongs to some Fock space if and only if its derivative belongs to a Fock space with a (possibly) different weight. Furthermore, we characterize the Borel measures $\mu$ for which a Fock space is continuously embedded in $L^q(\mu0)$ with $q>0$. We then illustrate the applicability of these results to the study of properties such as boundedness, compactness, Schatten class membership and the invariant subspaces of integration operators of Volterra type acting on Fock spaces. (joint work with Jose Angel Pelaez)
  • Functional Analysis Seminar
26 February 2013
17:00
to
18:16
Derek Kitson
Abstract
Laman's theorem characterises the minimally infinitesimally rigid frameworks in Euclidean space $\mathbb{R}^2$ in terms of $(2,3)$-tight graphs. An interesting problem is to determine whether analogous results hold for bar-joint frameworks in other normed linear spaces. In this talk we will show how this can be done for frameworks in the non-Euclidean spaces $(\mathbb{R}^2,\|\cdot\|_q)$ with $1\leq q\leq \infty$, $q\not=2$. This is joint work with Stephen Power.
  • Functional Analysis Seminar
12 February 2013
17:00
to
18:16
Charles Batty
Abstract

A very efficient way to obtain rates of energy decay for damped wave equations is to use operator semigroups to pass from resolvent estimates to energy estimates.  This is known to give the optimal results in cases when the resolvent estimates have simple forms such as being exactly polynomial ($|s|^\alpha$).  After reviewing that theory this talk will discuss cases when the resolvent estimates are slightly different, for example $|s|^\alpha/ \log |s|$ or $|s|^\alpha \log |s|$.

  • Functional Analysis Seminar
29 January 2013
17:00
to
18:16
Andrew Morris
Abstract
We prove that strongly continuous groups generated by first-order systems $D$ on Riemannian manifolds have finite propagation speed. The new direct proof for self-adjoint systems also provides a new approach to the weak Huygens' principle for second-order hyperbolic equations. The techniques are also combined with the resolvent approach to sectorial operators to obtain $L^2$ off-diagonal estimates for functions of $D$, which are the starting point for obtaining $L^p$ estimates for Riesz transforms on manifolds where the heat semigroup does not satisfy pointwise Guassian bounds. The two approaches are then combined via a Calder\'{o}n reproducing formula that allows for the analysing function to interact with $D$ through the holomorphic functional calculus whilst the synthesising function interacts with $D$ through the Fourier transform. This is joint work with P.~Auscher and A.~McIntosh.
  • Functional Analysis Seminar

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