Banach Spaces Seminar

The focus of the talk will be distributional inequalities for the volume of random convex sets. Typical models include convex hulls and Minkowski sums of line segments generated by independent random points. I will outline an approach to small-deviation estimates that makes use of rearrangement inequalities and tools from classical convexity such as intrinsic volumes and natural generalizations. (joint work with Grigoris Paouris)

As is well known, a complex $m\times m$ matrix $A$ is a commutator (i.e., there are matrices $B$ and $C$ of the same dimensions as $A$ such that $A=[B,C]=BC-CB$) if and only if $A$ has zero trace. In such a situation clearly $\|A\|\le 2\|B\|\|C\|$ where $\|D\|$ denotes the norm of $D$ as an operator from $\ell_2^m$ to itself. Does the converse hold? That is, if $A$ has zero trace are there $m\times m$ matrices $B$ and $C$ such that $A=[B,C]$ and $\|B\|\|C\|\le K\|A\|$ for some absolute constant $K$? If not, what is the behavior of the best $K$ as a function of $m$? I'll present some new insight, due to Johnson, Ozawa and myself concerning this question.

We investigate the question when does the trace formula: tr(T)=\sum\lambda_j(T) hold. Here \lambda_1(T),\lambda_2(T),... are all the eigenvalues of an operator T, with their multiplicities. Lidskii in 1957 proved that the formula holds for the Hilbert space. In the 1970's it was realized that the formula only can hold for spaces which are very close to the Hilbert space. Pisier in 1988 proved that the weak Hilbert spaces satisfy the formula. The weak Hilbert spaces are somewhat elusive and there are very few known examples of them. Bill Johnson and I have recently found a rather large clas of natural Banach spaces which satisfy the formula. The crucial role (like in Pisier's proof) is played by the quantity: G_n(X)= \sup{|det[<\phi_i, x_j>]^n_{i, j = 1}|: \phi_i \in B_{X*},x_j \in B_X}.

We plan to give an overview of results addressing the following type of problems: Given a Banach space X, we want to embed X into a Banach space Z which has some coordinate system with certain properties but is as close as possible to X.