## 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 Open image in new window . T(&) has a real spectrum without eigenvalues and its resolvent operator satisfies a first order growth (G_{i}). If T_{0} 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 T_{0} 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 T_{0}=T (&_{0}).

## Keywords

Banach Space Growth Condition Linear Transformation Dense Subset Resolvent Operator
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Download
to read the full article text

## References

- [1]R. G. Bartle,
*Spectral localization of operators in Banach spaces*, Math. Ann., vol. 153 (1964), pp. 261–269.CrossRefzbMATHMathSciNetGoogle Scholar - [2]N. Dunford,
*A survey of the theory of spectral operators*, Bull. Amer. Math. Soc., vol. 64 (1958), pp. 217–274.zbMATHMathSciNetCrossRefGoogle Scholar - [3]E. Hille andR. S. Phillips,
*Functional Analysis and semigroups*, Amer. Math. Soc. Colloquium Pubblications, volume XXXI, New York, 1957.Google Scholar - [4]D. Kocan,
*Spectral manifolds for a class of operators*. Illinois J. Math., Vol. 10 (1966), 605–622.zbMATHMathSciNetGoogle Scholar - [5]G. K. Leaf,
*A spectral theory for a class of linear operators*, Pacific J. Math., vol. 13 (1963), pp. 141–155.zbMATHMathSciNetGoogle Scholar - [6]E. R. Lorch,
*On a calculus of operators in reflexive vector spaces*, Trans. Amer. Math Soc., vol. 45 (1939), pp. 217–234.CrossRefzbMATHMathSciNetGoogle Scholar - [7]—— ——,
*The integral representation of weakly-almost periodic transformations in reflexive vector spaces*, Trans. Amer. Math. Soc., vol. 49 (1941), pp. 18–40.CrossRefzbMATHMathSciNetGoogle Scholar - [8]—— ——,
*Return to the selfadjoint transformation*, Acta. Sci. Szeged, vol. 12B (1950), pp. 137–144.MathSciNetGoogle Scholar - [9]—— ——,
*Spectral theory*, New York, Oxford U. Press, 1962.zbMATHGoogle Scholar - [10]M. H. Stone,
*Linear Transformations in Hilbert space*, Amer. Math. Soc. Colloquium Pubblications, vol. XV, New York, 1932.Google Scholar

## Copyright information

© Nicola Zanichelli Editore 1970