Forthcoming events in this series


Tue, 19 Nov 2013

17:00 - 18:30
C6

Maximal left ideals of operators acting on a Banach space

Tomasz Kania
(Lancaster)
Abstract

We address the following two questions regarding the maximal left ideals of the Banach algebras B(E) of bounded operators acting on an infinite-dimensional Banach space E:

i) Does B(E) always contain a maximal left ideal which is not finitely generated?

ii) Is every finitely-generated maximal left ideal of B(E) necessarily of the form {T\in B(E): Tx = 0}? for some non-zero vector x in E?

Since the two-sided ideal F(E) of finite-rank operators is not contained in any of the maximal left ideals mentioned above, a positive answer to the second question would imply a positive answer to the first.

Our main results are:

Question i) has a positive answer for most (possibly all) infinite-dimensional Banach spaces;

Question ii) has a positive answer if and only if no finitely-generated, maximal left ideal of B(E) contains F(E); the answer to Question ii) is positive for many, but not all, Banach spaces. We also make some remarks on a more general conjecture that a unital Banach algebra is finite-dimensional whenever all its maximal left ideals are finitely generated; this is true for C*-algebras.

This is based on a recent paper with H.G. Dales, T. Kochanek, P. Koszmider and N.J. Laustsen (Studia Mathematica, 2013) and work in progress with N.J. Laustsen.

Tue, 12 Nov 2013

17:00 - 18:12
C6

The heat equation in curved stripes

Martin Kolb
(Reading)
Abstract

We study the inuence of the intrinsic curvature on the large time behaviour of the heat equation in a tubular neighbourhood of an unbounded geodesic in a two-dimensional Riemannian manifold. Since we consider killing boundary conditions, there is always an exponential-type decay for the heat semigroup. We show that this exponential-type decay is slower for positively curved manifolds comparing to the flat case. As the main result, we establish a sharp extra polynomial-type decay for the heat semigroup on negatively curved manifolds comparing to the flat case. The proof employs the existence of Hardy-type inequalities for the Dirichlet Laplacian in the tubular neighbourhoods on negatively curved manifolds and the method of self-similar variables and weighted Sobolev spaces for the heat equation.

Tue, 22 Oct 2013

17:00 - 18:25
C6

The Dauns-Hofmann Theorem and tensor products of C*-algebras

David McConnell
(Trinity College Dublin)
Abstract

The problem of representing a (non-commutative) C*-algebra $A$ as the

algebra of sections of a bundle of C*-algebras over a suitable base

space may be viewed as that of finding a non-commutative Gelfand-Naimark

theorem. The space $\mathrm{Prim}(A)$ of primitive ideals of $A$, with

its hull-kernel topology, arises as a natural candidate for the base

space. One major obstacle is that $\mathrm{Prim}(A)$ is rarely

sufficiently well-behaved as a topological space for this purpose. A

theorem of Dauns and Hofmann shows that any C*-algebra $A$ may be

represented as the section algebra of a C*-bundle over the complete

regularisation of $\mathrm{Prim}(A)$, which is identified in a natural

way with a space of ideals known as the Glimm ideals of $A$, denoted

$\mathrm{Glimm}(A)$.

In the case of the minimal tensor product $A \otimes B$ of two

C*-algebras $A$ and $B$, we show how $\mathrm{Glimm}(A \otimes B)$ may

be constructed in terms of $\mathrm{Glimm}(A)$ and $\mathrm{Glimm} (B)$.

As a consequence, we describe the associated C*-bundle representation of

$A \otimes B$ over this space, and discuss properties of this bundle

where exactness of $A$ plays a decisive role.

Tue, 11 Jun 2013

17:00 - 18:15
L3

The construction of quantum dynamical semigroups by way of non-commutative Markov processes

Alex Belton
(Lancaster)
Abstract

Although generators of strongly continuous semigroups of contractions

on Banach spaces are characterised by the Hille-Yosida theorem, in

practice it can be difficult to verify that this theorem's hypotheses

are satisfied. In this talk, it will be shown how to construct certain

quantum Markov semigroups (strongly continuous semigroups of

contractions on C* algebras) by realising them as expectation

semigroups of non-commutative Markov processes; the extra structure

possessed by such processes is sufficient to avoid the need to use

Hille and Yosida's result.

Tue, 14 May 2013

17:00 - 18:07
L3

The Dirichlet-to-Neumann operator on rough domains

Tom ter Elst
(Auckland)
Abstract

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 May 2013

17:00 - 18:10
L1

Analytical aspects of isospectral drums

James Kennedy
(Ulm)
Abstract

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, 23 Apr 2013

17:00 - 18:08
L3

On the representation of tight functionals as integrals

David Edwards
(Oxford)
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.

Tue, 05 Mar 2013

17:00 - 18:16
L3

Carleson embeddings and integration operators of Volterra type on Fock spaces

Olivia Constantin
(Kent)
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)

Tue, 26 Feb 2013

17:00 - 18:16
L3

A Laman theorem for non-Euclidean bar-joint frameworks.

Derek Kitson
(Lancaster)
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.

Tue, 12 Feb 2013

17:00 - 18:16
L3

Fine scales of decay rates of operator semigroups

Charles Batty
(Oxford)
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 $|s|^\alpha \log |s|$.

Tue, 29 Jan 2013

17:00 - 18:16
L3

Huygens' Principle for Hyperbolic Equations and $L^p$ Estimates for Riesz Transforms on Manifolds via First-Order Systems

Andrew Morris
(Oxford)
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.

Tue, 30 Oct 2012

17:00 - 18:23
L3

Spectral problems for semigroups and asymptotically disappearing solutions

Vesselin Petkov
(Bordeaux)
Abstract

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

Tue, 23 Oct 2012

17:00 - 18:23
L3

Joint numerical radius

Vladimir Muller
(Czech Academy of Sciences)
Abstract

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 May 2012

17:00 - 18:10
L3

A TRACE FORMULA AND STABILITY OF SQUARE ROOT DOMAINS FOR NON-SELF-ADJOINT OPERATORS

Fritz Gesztesy
(Missouri)
Abstract

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 dierential operators in

divergence form on bounded Lipschitz domains.

This is based on various joint work with S. Hofmann, R. Nichols, and M. Zinchenko.