A functional calculus for unbounded generalized scalar operators on Banach spaces (Q618690)

Let \(T=(T_1,\dots,T_k)\) be a \(k\)-tuple of closed densely defined operators on a Banach space that verifies certain conditions. Let \(X\) be a reflexive Banach space and let \(\alpha \geq 0\). The authors prove that \[ f \in F_{\alpha} \mapsto \Phi(f) = f(T) = \frac{1}{(2 \pi)^{k/2}} \int_{R^k} e^{-it.T}\, d\mu_{\widehat{f}} \in B(X) ,\tag{1} \] where the measure \(\mu_{\widehat{f}}\), the Fourier transform of the function \(f\), is a continuous homomorphism, \(F_{\alpha}\) is a certain algebra, and the integral exists as a weak integral. That is, formula (1) defines an ultraweak functional calculus (this notion is defined in the paper).