A complement to spectral mapping theorems for  C  0 ‐semigroups | Zendy

Król Sebastian | Zendy

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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.

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