The Semi-simplicity Manifold on Arbitrary Banach Spaces

For an arbitrary linear (possibly unbounded) operator A on a Banach space, with real spectrum, we construct a maximal continuously embedded Banach subspace on which this operator has a Cℓ(R) functional calculus. We call this subspace, Z, the semi-simplicity manifold for A. When the original Banach space does not contain a copy of c0, the restriction of A to Z is a spectral operator of scalar type. We construct a functional calculus, f ↦ f(A |Z), from C(R) into the space of closed, densely defined operators on Z; when X does not contain a copy of c0, this map is defined for arbitrary Borel measurable f. We also construct continuously embedded Banach subspaces on which the Fourier transform and the Hilbert transform are spectral operators of scalar type.