Past Functional Analysis Seminar

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