Strongly continuous spectral families

Summary We define a strongly continuous family & of bounded projections E(t), t real, on a Banach space X and show that & generates a densely defined closed linear transformation in X given by . T(&) has a real spectrum without eigenvalues and its resolvent operator satisfies a first order growth (G_i). If T₀ is a given closed linear trasformation defined a dense subset of X which has a purely continuous real spectrum and a resolvent operator satisfying the first order growth condition (G_i) then T₀ has a ? resolution of the identity ? &₀ consisting of closed projections E(t) in X. We show that if &₀ is also strongly continuous then T₀=T (&₀). Dedicated to the sixtieth birthday of Professor Edgar. R. Lorch