Past Functional Analysis Seminar

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

In this talk I will describe recent work by myself and others (E.B. Davies (KCL), M. Lindner (Chemnitz), S. Roch (Darmstadt)) on the spectrum and essential spectrum of bi-infinite and semi-infinite (not necessarily self-adjoin) tri-diagonal random operators, and the implications of these results for the spectra of associated random matrices, and for the finite section method for infinite tri-diagonal systems. A main tool will be limit operator methods, as described in Chandler-Wilde and Lindner, Memoirs AMS, 2011), supplemented by certain symmetry arguments including a Coburn lemma for random matrices.

The talk presents a collection of results about mapping properties

of the Hardy-Steklov operator

$(Hf)(x)=\int_{a(x)}^{b(x)} f(y) dy$ in weighted Lebesgue spaces on the

semi-axis. In particular, the explicit boundedness and compactness criteria

for the operator are given and a number of applications are obtained. A

part of the results is based on a joint paper with Prof. V.D. Stepanov

I will briefly discuss boundedness and compactness of Toeplitz operators

on Bergman spaces and then describe their essential spectra for several

symbol classes (e.g., the Douglas algebra, VMO and BMO type spaces,

matrix-valued symbols). I will also list some open problems related to

boundedness, compactness and Fredholmness.

One of the cores in modern probability theory is the stochastic integral introduced by K.

Ito in the 1940s. Due to the randomness and the irregularity of typical stochastic

integrators (such as the Wiener process) one can not follow a classical approach as in

calculus to define the stochastic integral.

For Hilbert spaces Ito's theory of stochastic integration in finite

dimensions can be generalised. There are several even quite early attempts to tackle

stochastic integration in more general spaces such as Banach spaces but none of them

provides the generality and powerful tool as the theory in Hilbert spaces.

In this talk, we begin with introducing the stochastic integral in Hilbert spaces based

on the classical theory and with explaining the restriction of this approach to Hilbert

spaces. We tackle the problem of stochastic integration in Banach spaces by introducing

a stochastic version of a Pettis integral. In the case of a Wiener process as an integrator,

the stochastic Pettis integrability of a function is related to the extensively studied class of

$\gamma$-radonifying operators. Surprisingly, it turns out that for more general integrators

which are non-Gaussian and discontinuous (Levy processes) such a relation can still be

established but with another subclass of radonifying operators.

This will be mainly a survey talk covering recently-resolved conjectures of Polya and Wiman for entire functions, and progress on extensions to meromorphic functions