The Weyl calculus and a Cayley–Hamilton theorem for pairs of selfadjoint matrices

AbstractThe Weyl calculus WA for a pair of selfadjoint matrices A=(A1,A2) is a construction (originally devised by H. Weyl and based on the theory of Fourier transforms) which associates a matrix WA(f) to each smooth function f defined on R2. The association f↦WA(f) is linear but typically not multiplicative. For a single selfadjoint matrix B, the matrix WB(f) is also defined and is known to coincide with the matrix f(B) as given by the classical spectral theorem. In recent years it has been shown that certain analytic, geometric and topological properties of WA and/or the support of WA (an appropriately defined subset of R2) have strong implications for the relationship between A1 and A2. The aim of this note is to contribute an additional (and rather remarkable) property of WA, of a distinctly different nature (i.e. an algebraic condition). Namely, if cA denotes the joint characteristic polynomial of the pair A, i.e. the function λ↦det[(A1−λ1I)2+(A2−λ2I)2] for λ∈R2, then A1A2=A2A1 if and only if WA vanishes on the single polynomial function cA. The requirement WA(cA)=0 can be interpreted as a “vector analogue” of the Cayley–Hamilton theorem: our result states that this is satisfied if and only if A1 and A2 commute