Strongly continuous spectral families
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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 Open image in new window . 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).
KeywordsBanach Space Growth Condition Linear Transformation Dense Subset Resolvent Operator
- E. Hille andR. S. Phillips,Functional Analysis and semigroups, Amer. Math. Soc. Colloquium Pubblications, volume XXXI, New York, 1957.Google Scholar
- M. H. Stone,Linear Transformations in Hilbert space, Amer. Math. Soc. Colloquium Pubblications, vol. XV, New York, 1932.Google Scholar