On the Weyl functional calculus

AbstractThe operator valued distribution T(A) on Rn associated with an n-tuple of self-adjoint operators (A) which was discussed in “The Weyl Functional Calculus” is further studied. The results are: 1.(1) The support of the distribution T(A) is connected in Rn unless some nontrivial orthogonal projection commutes with all the operators in the n-tuple (A).2.(2) In the special case of finite dimensional Hilbert space, T(A) cannot be represented as a measure unless all the operators in the n-tuple (A) commute.The first result is obtained by studying the propagation of an operator differential equation of which T(A) is a solution.The second result is obtained by geometric study of the numerical range of an n-tuple of operators.The results are used to contrast the Weyl calculus for a pair of self-adjoint operators with the closely related Riesz calculus for a bounded operator