## Some Results in the Hyperinvariant Subspace Problem and Free Probability

##### Abstract

This dissertation consists of three more or less independent projects. In the first
project, we find the microstates free entropy dimension of a large class of L1[0; 1]{
circular operators, in the presence of a generator of the diagonal subalgebra.
In the second one, for each sequence {cn}n in l1(N), we de fine an operator A in
the hyper finite II1-factor R. We prove that these operators are quasinilpotent and
they generate the whole hyper finite II1-factor. We show that they have non-trivial,
closed, invariant subspaces affiliated to the von Neumann algebra, and we provide
enough evidence to suggest that these operators are interesting for the hyperinvariant
subspace problem. We also present some of their properties. In particular, we
show that the real and imaginary part of A are equally distributed, and we find a
combinatorial formula as well as an analytical way to compute their moments. We
present a combinatorial way of computing the moments of A*A.
Finally, let fTkg1k =1 be a family of *-free identically distributed operators in a
finite von Neumann algebra. In this paper, we prove a multiplicative version of the
Free Central Limit Theorem. More precisely, let Bn = T*1T*2...T*nTn...T2T1 then
Bn is a positive operator and B1=2n
n converges in distribution to an operator A. We
completely determine the probability distribution v of A from the distribution u of
jTj2. This gives us a natural map G : M M with u G(u) = v. We study
how this map behaves with respect to additive and multiplicative free convolution. As an interesting consequence of our results, we illustrate the relation between the
probability distribution v and the distribution of the Lyapunov exponents for the
sequence fTkg1k=1 introduced by Vladismir Kargin.