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 (Gi). 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 (Gi) 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).
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Dedicated to the sixtieth birthday of Professor Edgar. R. Lorch
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Kocan, D. Strongly continuous spectral families. Annali di Matematica 86, 31–42 (1970). https://doi.org/10.1007/BF02415706
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DOI: https://doi.org/10.1007/BF02415706