Nonanalytic functional calculi and spectral maximal spaces

Curtis, Philip C.; Neumann, Michael

doi:10.2140/pjm.1989.137.65

Cited by 6 publications

(2 citation statements)

References 19 publications

(16 reference statements)

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“…Let us point out that the best possible exponent m ¼ 2 in [11] is attained when A admits a CðRÞ-calculus. By Kantorovitz [26, Lemma 2.2], the spectrum sðAÞ is then a subset of R. The following is an example of an operator A 2 LðX Þ such that sðAÞ & R; iA generates a group of isometries (in particular a trivially asymptotically stable semigroup), but A does not admit a CðRÞ-calculus.…”

Section: Proof Of ð9þ ) ð3þ ð1þmentioning

confidence: 99%

Strong Stability of Bounded Evolution Families and Semigroups

Batty¹,

Chill²,

Tomilov

2002

Journal of Functional Analysis

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We prove several characterizations of strong stability of uniformly bounded evolution families ðUðt; sÞÞ t5s50 of bounded operators on a Banach space X , i.e. we characterize the property lim t!1 jjUðt; sÞxjj ¼ 0 for all s50 and all x 2 X . These results are connected to the asymptotic stability of the well-posed linear nonautonomous Cauchy problem ' u uðtÞ ¼ AðtÞuðtÞ; t5s50; uðsÞ ¼ x;x 2 X : ( In the autonomous case, i.e. when Uðt; sÞ ¼ Tðt À sÞ for some C 0 -semigroup ðTðtÞÞ t50 , we present, in addition, a range condition on the generator A of ðTðtÞÞ t50 which is sufficient for strong stability. This condition is more general than the condition in the ABLV-Theorem involving countability of the imaginary part of the spectrum of A. # 2002 Elsevier Science (USA)

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Section: Proof Of ð9þ ) ð3þ ð1þmentioning

confidence: 99%

Strong Stability of Bounded Evolution Families and Semigroups

Batty¹,

Chill²,

Tomilov

2002

Journal of Functional Analysis

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“…for all sufficiently large integers p and all closed sets F ⊆ C; see also [4], [7], and [10, 1.4.14, 1. …”

Section: Introduction and Basic Definitionsmentioning

confidence: 99%

Spectral subspaces of subscalar and related operators

Miller¹,

Miller²,

Neumann³

2003

Proc. Amer. Math. Soc.

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Abstract. For a bounded linear operator T ∈ L(X) on a complex Banach space X and a closed subset F of the complex plane C, this note deals with algebraic representations of the corresponding analytic spectral subspace X T (F ) from local spectral theory. If T is the restriction of a generalized scalar operator to a closed invariant subspace, then it is shown thatp X for all sufficiently large integers p, whereMoreover, for a wide class of operators T that satisfy growth conditions of polynomial or Beurling type, it is shown that X

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Commutative Banach algebras and decomposable operators

Neumann

1992

Monatshefte für Mathematik

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Nonanalytic functional calculi and spectral maximal spaces

Cited by 6 publications

References 19 publications

Strong Stability of Bounded Evolution Families and Semigroups

Strong Stability of Bounded Evolution Families and Semigroups

Spectral subspaces of subscalar and related operators

Commutative Banach algebras and decomposable operators

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