Past Functional Analysis Seminar

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).

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.

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)

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.

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 $|s|^\alpha \log |s|$.

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.

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

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 $w(T)\ge \|T\|/2$.

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.

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.

A theorem of R. Nevanlinna from 1922 characterizes the Cauchy

transforms of finite positive measures on the real line as the functions in

the Pick class that satisfy a certain growth condition on the real axis; this

result is important in the spectral theory of self-adjoint operators.

(The Pick class is the set of analytic functions in the upper half-plane

$\Pi$ with non-negative imaginary part). I will describe a higher-dimensional

analogue of Nevanlinna's theorem. The $n$-variable Pick class is defined to

be the set of analytic functions on the polyhalfplane $\Pi^n$ with non-negative

imaginary part; we obtain four different representation formulae for functions

in the $n$-variable Pick class in terms of the ``structured resolvent" of a

densely defined self-adjoint operator. Structured resolvents are analytic

operator-valued functions on the polyhalfplane with properties analogous to those of the

familiar resolvent of a self-adjoint operator. The types of representation that a

function admits are determined by the growth of the function on the imaginary polyaxis $(i\R)^n$.

Rordam (09.30-10.15): continued from Friday

Musat (10.30-11,30; 11.45-12.30): Factorizable completely positive maps and quantum information

theory

Pisier (14.00-15.00, 15.15-16.00)

Unconditionality of Martingale Differences (UMD) for Banach and Operator Spaces

Rordam (16.30-17.30, continued Saturday)

Kirchberg algebras arising from groups acting on compact and locally compact spaces