Spectral representations for unbounded operators with real spectrum (Q1095378)

The semisimplicity manifold Z of an arbitrary unbounded operator T with real spectrum acting in a reflexive Banach space X is defined. For all \(x\in D(T| Z)\), one has \(Tx=\int _{R}sE(ds)x\), where E is a spectral measure on Z. For suitable \(x\in Z\), the abstract Cauchy problem \((du/dt)=iTu\), \(u(0)=x\) is shown to have the solution \(u(t)=G(t)x=\int _{R}e^{its}E(ds)x\). Burnap and Zweifel's concept of U-spectrality and many of their results are put in the context of the semisimplicity manifold, yielding certain natural extensions.