Differentiation of an operator valued measure in a Banach rigging (Q1119872)

Let A be a self-adjoint operator on a Hilbert space \(H_ 0\), and E(.) its resolution of the identity. Then impose a condition on \(H_ 0\) known as Banach equipment, roughly, \(B_-\supset H_ 0\supset B_+\) where \(B_+\) and \(B_-\) are Banach spaces ``close'' to \(H_ 0\). The author shows that under certain conditions on A involving Lebesgue measure \(\lambda\), there is a \(\sigma\)-additive scalar measure \(\mu\) (\(\lambda)\) and there is a continuous non-negative operator \(\Psi\) (\(\lambda)\) from \(B_+\) into \(B_-\) such that for \(u,v\in B_+\) we have \[ (E(\Omega)u,v)_ 0=\int_{\Omega}(\Psi (\lambda)u,v)_ 0 d\mu (\lambda), \] where \((.,.)_ 0\) denotes the inner product in \(H_ 0\).