Abstract

The Weyl calculus for a pair A = (A₁, A₂) of self-adjoint (n × n)-matrices, due to H. Weyl, associates a matrix W _A (ƒ) to each smooth function ƒ defined on R² in a linear but typically not multiplicative way. Letting c _A(λ) := det((A₁ - λ₁I)² + (A₂ - λ₂I)²) for λ ∈ R² denote the joint characteristic polynomial of the pair A, it is known, for n ≤ 3, that A₁A₂ = A₂A₁ if and only if W_A(c_A) = 0. It is an open question whether this is still true for n ≥ 4. Our aim here is to pursue two new approaches: the role of the canonical order structure for self-adjoint matrices; and topological invariants arising from continuity properties of the non-linear map (A, ƒ) → W_A(ƒ).