COMMUTATIVITY OF COMPACT SELF-ADJOINT OPERATORS

The relationship between the joint spectrum gamma(A) of an n-tuple A = (A1,...,A(n)) of selfadjoint operators and the support of the corresponding Weyl calculus T(A) : f bar arrow pointing right f(A) is discussed. It is shown that one always has gamma(A) subset-of supp(T(A)). Moreover, when the operators are compact, equality occurs if and only if the operators A(j) mutually commute. In the non-commuting case the equality fails badly: While gamma(A) is countable, supp(T(A)) has to be an uncountable set. An example is given showing that, for non-compact operators, coincidence of gamma(A) and supp(T(A)) no longer implies commutativity of the set {A(i)}.