Feynman's operational calculi: spectral theory for noncommuting self-adjoint operators
Let \((A_1,\dots,A_n)\) be an \(n\)-tuple of linear bounded operators acting on a Banach space \(X\). The continuous probability measures \(\mu_1,\dots,\mu_n\), defined on the Borel class of \([0,1]\), determine an operational calculus from a commutative Banach algebra \(\mathbb{D}(A_1,\dots,A_n)\) (the disentangling algebra of analytic functions) into the Banach algebra \(L(X)\) of all the operators on \(X\). In this paper, it is proved the following result: an \(n\)-tuple \((A_1,\dots,A_n)\) of bounded selfadjoint operators on a Hilbert space \(H\) is of Paley-Wiener type \((0,r,\mu)\), with \(r=(\|A_1\|^2 + \cdots + \|A_n\|^2)^{1/2}\), for any \(n\)-tuple \(\mu_1,\dots,\mu_n\), a property related with the above functional calculus.
scientific article; zbMATH DE number 5556069