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A complement to spectral mapping theorems for C 0 ‐semigroups
Author(s) -
Król Sebastian
Publication year - 2011
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/bdq076
Subject(s) - mathematics , complement (music) , semigroup , banach space , generator (circuit theory) , pure mathematics , range (aeronautics) , spectrum (functional analysis) , discrete mathematics , operator (biology) , biochemistry , chemistry , complementation , gene , phenotype , power (physics) , physics , materials science , repressor , quantum mechanics , transcription factor , composite material
We obtain a complement to the classical spectral mapping (inclusion) theorems for C 0 ‐semigroups. More precisely, we show that if ( T ( t )) t ⩾0 is a C 0 ‐semigroup on a complex Banach space X with generator A and if, for fixed k ∈ ℕ and t > 0, the operator ( T ( t )− I ) k has closed range, then the range of A k is also closed. In particular, for k = 1 this implies the validity of the following version of the spectral mapping (inclusion) theoreme t σ ℛ ( A ) ⊂ σ ℛ ( T ( t ) ) ,t > 0 ,where σ ℛ ( A ) := { λ ∈ℂ: the range of λ − A is not closed}. Our results are, in a sense, optimal.