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A Remark on Contraction Semigroups on Banach Spaces
Author(s) -
Lin PieKee
Publication year - 1995
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/27.2.169
Subject(s) - mathematics , banach space , contraction (grammar) , hilbert space , eigenvalues and eigenvectors , unit sphere , pure mathematics , semigroup , regular polygon , discrete mathematics , combinatorics , mathematical analysis , geometry , physics , medicine , quantum mechanics
Let X be a complex Banach space and let J : X → X * be a duality section on X (that is, 〈 x , J ( x )〉=∥ J ( x )∥∥ x ∥=∥ J ( x )∥ 2 )=∥ x ∥ 2 ). For any unit vector x and any ( C 0 ) contraction semigroup T ={e tA : t ⩾0}, Goldstein proved that if X is a Hilbert space and ∣〈 T ( t ) x , j ( x )〉∣→1 as t →∞, then x is an eigenvector of A corresponding to a purel imaginary eigenvalue. In this article, we prove that a similar result holds if X is a strictly convex complex Banach space.