Commutativity of compact selfadjoint operators

The relationship between the joint spectrum γ(A) of an n-tuple ${\displaystyle A=(A_1,...,A_n)}$ of selfadjoint operators and the support of the corresponding Weyl calculus T(A) : f ↦ f(A) is discussed. It is shown that one always has γ(A) ⊂ supp (T(A)). Moreover, when the operators are compact, equality occurs if and only if the operators ${\displaystyle A_j}$ mutually commute. In the non-commuting case the equality fails badly: While γ(A) is countable, supp(T(A)) has to be an uncountable set. An example is given showing that, for non-compact operators, coincidence of γ(A) and supp (T(A)) no longer implies commutativity of the set ${\displaystyle \left\{A_i\right\}}$ .