Annali di Matematica Pura ed Applicata

, Volume 86, Issue 1, pp 31–42 | Cite as

Strongly continuous spectral families

  • Daniel Kocan


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).


Banach Space Growth Condition Linear Transformation Dense Subset Resolvent Operator 
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  1. [1]
    R. G. Bartle,Spectral localization of operators in Banach spaces, Math. Ann., vol. 153 (1964), pp. 261–269.CrossRefzbMATHMathSciNetGoogle Scholar
  2. [2]
    N. Dunford,A survey of the theory of spectral operators, Bull. Amer. Math. Soc., vol. 64 (1958), pp. 217–274.zbMATHMathSciNetCrossRefGoogle Scholar
  3. [3]
    E. Hille andR. S. Phillips,Functional Analysis and semigroups, Amer. Math. Soc. Colloquium Pubblications, volume XXXI, New York, 1957.Google Scholar
  4. [4]
    D. Kocan,Spectral manifolds for a class of operators. Illinois J. Math., Vol. 10 (1966), 605–622.zbMATHMathSciNetGoogle Scholar
  5. [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. [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. [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. [8]
    —— ——,Return to the selfadjoint transformation, Acta. Sci. Szeged, vol. 12B (1950), pp. 137–144.MathSciNetGoogle Scholar
  9. [9]
    —— ——,Spectral theory, New York, Oxford U. Press, 1962.zbMATHGoogle Scholar
  10. [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

Authors and Affiliations

  • Daniel Kocan
    • 1
  1. 1.HobokenU.S.A.

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