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

The Weyl calculus W-A for a pair of selfadjoint matrices A = (A(1), A(2)) is a construction (originally devised by H. Weyl and based on the theory of Fourier transforms) which associates a matrix W-A(f) to each smooth function f defined on R-2. The association f bar right arrow W-A(f) is linear but typically not multiplicative. For a single selfadjoint matrix B, the matrix W-B(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 W-A and/or the support of W-A (an appropriately defined subset of R-2) have strong implications for the relationship between A(1) and A(2). 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 c(A) denotes the joint characteristic polynomial of the pair A, i.e. the function lambda bar right arrow det[(A(1) - lambda I-1)(2) + (A(2) - lambda I-2)(2)] for lambda is an element of R-2, then A(1)A(2) = A(2)A(1) if and only if W-A vanishes on the single polynomial function c(A). The requirement W-A(c(A)) = 0 can be interpreted as a "vector analogue" of the Cayley-Hamilton theorem: our result states that this is satisfied if and only if A(1) and A(2) commute. (C) 2000 Elsevier Science Inc. All rights reserved.