ILAS2016 — 11–15 July 2016 — KU Leuven, Belgium

20th Conference of the International Linear Algebra Society (ILAS)

20th ILAS Conference

In minisymposium: Graph Theory and Linear Algebra

Thu 10:30–11:00, Auditorium Max Weber
Symmetric seminorms and the Markov property
Zoltan Leka (Royal Holloway, University of London)

Recently M.A. Rieffel proved that the (non-commutative) standard deviation $\sigma$ is a strongly Leibniz seminorm; that is, the Leibniz inequality $\sigma(fg) \leq \sigma(f)\|g\| + \sigma(g)\|f\|$ is satisfied with bounded functions on (non-commutative) probability spaces, where the norm is the supremum norm. Additionally, a family of related inequalities holds with real-valued Lipschitz functions which covers the 'strong' part of the result.

It seems to be a natural question whether seminorms determined by higher order central moments possess the aforementioned properties or not. During the talk we shall focus on the case of real functions defined on finite sets. Whilst the $\ell^2$ case can be discussed relying upon the theory of Dirichlet forms and resistance networks, the $\ell^p$ case reflects severe nonlinearity. To get an efficient and friendly approach, we need to replace the $\ell^p$ norms with symmetric norms. We show that all symmetric central seminorms on $\mathbb{R}^n$ satisfy the Leibniz property and the so-called Markov property as well (which is much stronger than the previous one). The proofs are based on matrix constructions, results in majorization theory and interpolation of Laplacians on a certain subspace of codimension one. Furthermore, we shall provide a derivation which drives the results behind the scenes.