Past Seminars › Seminar Series

Functional Analysis Seminar (past)

Tue, 14/05
17:00 		Tom ter Elst (Auckland) Functional Analysis Seminar Add to calendar L3
The Dirichlet-to-Neumann operator on rough domains
We consider a bounded connected open set $ \Omega \subset {\rm R}^d $ whose boundary $ \Gamma $ has a finite $ (d-1) $-dimensional Hausdorff measure. Then we define the Dirichlet-to-Neumann operator $ D_0 $ on $ L_2(\Gamma) $ by form methods. The operator $ -D_0 $ is self-adjoint and generates a contractive $ C_0 $-semigroup $ S = (S_t)_{t > 0} $ on $ L_2(\Gamma) $. We show that the asymptotic behaviour of $ S_t $ as $ t \to \infty $ is related to properties of the trace of functions in $ H^1(\Omega) $ which $ \Omega $ may or may not have. We also show that they are related to the essential spectrum of the Dirichlet-to-Neumann operator. The talk is based on a joint work with W. Arendt (Ulm).
Thu, 09/05
17:00 		James Kennedy (Ulm) Functional Analysis Seminar Add to calendar L1
Analytical aspects of isospectral drums
Almost 50 years ago, Kac posed the now-famous question `Can one hear the shape of a drum?', that is, if two planar domains are isospectral with respect to the Dirichlet (or Neumann) Laplacian, must they necessarily be congruent? This question was answered in the negative about 20 years ago with the construction of pairs of polygonal domains with special group-theoretically motivated symmetries, which are simultaneously Dirichlet and Neumann isospectral. We wish to revisit these examples from an analytical perspective, recasting the arguments in terms of elliptic forms and intertwining operators. This allows us to prove in particular that the isospectrality property holds for a far more general class of elliptic operators than the Laplacian, as it depends purely on what the intertwining operator does to the form domains. We can also show that the same type of intertwining operator cannot intertwine the Robin Laplacian on such domains. This is joint work with Wolfgang Arendt (Ulm) and Tom ter Elst (Auckland).
Tue, 07/05
17:00 		Garth Dales (Lancaster) Functional Analysis Seminar Add to calendar L3
C(X) as a bidual space
Tue, 23/04
17:00 		David Edwards (Oxford) Functional Analysis Seminar Add to calendar L3
On the representation of tight functionals as integrals
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.
Tue, 05/03
17:00 		Olivia Constantin (Kent) Functional Analysis Seminar Add to calendar L3
Carleson embeddings and integration operators of Volterra type on Fock spaces
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)
Tue, 26/02
17:00 		Derek Kitson (Lancaster) Functional Analysis Seminar Add to calendar L3
A Laman theorem for non-Euclidean bar-joint frameworks.
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.
Tue, 12/02
17:00 		Charles Batty (Oxford) Functional Analysis Seminar Add to calendar L3
Fine scales of decay rates of operator semigroups

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 $|s|^\alpha \log |s|$.
Tue, 29/01
17:00 		Andrew Morris (Oxford) Functional Analysis Seminar Add to calendar L3
Huygens' Principle for Hyperbolic Equations and $ L^p $ Estimates for Riesz Transforms on Manifolds via First-Order Systems
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 Calderón 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.
Tue, 15/01
17:00 		Chris Heunen (Oxford) Functional Analysis Seminar Add to calendar L3
Diagonalizing matrices over AW*-algebras
Tue, 27/11/2012
17:00 		David Preiss (Warwick) Functional Analysis Seminar Add to calendar L3
Differentiability problems
Tue, 20/11/2012
17:00 		Pablo Shmerkin (Surrey) Functional Analysis Seminar Add to calendar L3
Normal numbers and fractal measures
Tue, 13/11/2012
17:00 		Yuri Safarov (KCL) Functional Analysis Seminar Add to calendar L3
ALMOST COMMUTING OPERATORS
Tue, 30/10/2012
17:00 		Vesselin Petkov (Bordeaux) Functional Analysis Seminar Add to calendar L3
Spectral problems for semigroups and asymptotically disappearing solutions
We study symmetric systems with dissipative boundary conditions. The solutions of the mixed problems for such systems are given by a contraction semigroup $ V(t)f = e^{tG_b}f,\: t \geq 0 $. The solutions $ u = e^{tG_b}f $ with eigenfunctions $ f $ of the generator $ G_b $ with eigenvalues $ \lambda,\: \Re \lambda < 0, $ are called asymptotically disappearing (ADS). If we have (ADS), the wave operators are not complete and the inverse back-scattering problems become complicated. We examine the spectrum of the generator $ G_b $ and we show that this spectrum in the open half plane $ \Re \lambda < 0 $ is formed by isolated eigenvalues with finite multiplicity. The existence of (ADS) is a difficult problem. We establish the existence of (ADS) for the Maxwell system in the exterior of a sphere. We will discuss some other applications related to the existence of (ADS).
Tue, 23/10/2012
17:00 		Vladimir Muller (Czech Academy of Sciences) Functional Analysis Seminar Add to calendar L3
Joint numerical radius
Let $ T_1,\dots,T_n $ be bounded linear operators on a complex Hilbert space $ H $. We study the question whether it is possible to find a unit vector $ x\in H $ such that $ |\langle T_jx, x\rangle| $ is large for all $ j $. Thus we are looking for a generalization of the well-known fact for $ n = 1 $ that the numerical radius $ w(T) $ of a single operator T satisfies $ w(T)\ge \|T\|/2 $.
Tue, 15/05/2012
17:00 		Fritz Gesztesy (Missouri) Functional Analysis Seminar Add to calendar L3
A TRACE FORMULA AND STABILITY OF SQUARE ROOT DOMAINS FOR NON-SELF-ADJOINT OPERATORS
We extend the classical trace formula connecting the trace of resolvent dif- ferences of two (not necessarily self-adjoint) operators A and A0 with the logarithmic derivative of the associated perturbation determinant from the standard case, where A and A0 have comparable domains (i.e., one contains the other) to the case where their square root domains are comparable. This is done for a class of positive-type operators A, A0. We then prove an abstract result that permits to compare square root domains and apply this to the concrete case of 2nd order elliptic partial di erential operators in divergence form on bounded Lipschitz domains. This is based on various joint work with S. Hofmann, R. Nichols, and M. Zinchenko.
Tue, 15/05/2012
09:30 		Jan van Neerven (Delft University of Technology) Functional Analysis Seminar Add to calendar L3
The stochastic Weiss conjecture
The stochastic Weiss conjecture is the statement that for linear stochastic evolution equations governed by a linear operator $ A $ and driven by a Brownian motion, a necessary and sufficient condition for the existence of an invariant measure can be given in terms of the operators $ \sqrt{\lambda}(\lambda-A)^{-1} $. Such a condition is presented in the special case where $ -A $ admits a bounded $ H^\infty $-calculus of angle less than $ \pi/2 $. This is joint work with Jamil Abreu and Bernhard Haak.

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