A multivariable Cayley--Hamilton theorem
Speaker: M. Mathieu
Date: Tue 4th September 2007
The Weyl calculus for a pair A = (A1,A2) of selfadjoint n × n matrices, due to H. Weyl, associates a matrix WA(f) to each smooth function f defined on 2 in a linear but typically not multiplicative way.
Letting cA(λ) = det((A1 - λ1)2 + (A2 - λ2)2) for λ = (λ1 ,λ2) alt="?" />2 denote the joint characteristic polynomial of the pair A it is known, for n ≤ 3, that A 1A2 = A2A1 if and only if WA(cA) = 0. It is an open problem whether this is still true for n > 3. We shall discuss two new approaches to this problem: the role of the canonical order structure for selfadjoint matrices; and topological invariants arising from continuity properties of the non-linear map (f,A) ?→ W A(f). This is joint work with W. Ricker, Eichstatt, Germany to be published in Math. Proc . Royal Ir. Acad.
(This talk is part of the IMS September Meeting 2007 series.)