Rank-1 perturbations of cosine functions and semigroups
Let A be the generator of a cosine function on a Banach space X. In many cases, for example if X is a UMD-space, A + B generates a cosine function for each B ∈ L (D ((ω - A)1 / 2), X). If A is unbounded and frac(1, 2) < γ ≤ 1, then we show that there exists a rank-1 operator B ∈ L (D ((ω - A)...
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| Format: | Journal article |
| Language: | English |
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Elsevier
2006
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| _version_ | 1797065027938680832 |
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| author | Batty, C Arendt, W |
| author_facet | Batty, C Arendt, W |
| author_sort | Batty, C |
| collection | OXFORD |
| description | Let A be the generator of a cosine function on a Banach space X. In many cases, for example if X is a UMD-space, A + B generates a cosine function for each B ∈ L (D ((ω - A)1 / 2), X). If A is unbounded and frac(1, 2) < γ ≤ 1, then we show that there exists a rank-1 operator B ∈ L (D ((ω - A)γ), X) such that A + B does not generate a cosine function. The proof depends on a modification of a Baire argument due to Desch and Schappacher. It also allows us to prove the following. If A + B generates a distribution semigroup for each operator B ∈ L (D (A), X) of rank-1, then A generates a holomorphic C0-semigroup. If A + B generates a C0-semigroup for each operator B ∈ L (D ((ω - A)γ), X) of rank-1 where 0 < γ < 1, then the semigroup T generated by A is differentiable and {norm of matrix} T′ (t) {norm of matrix} = O (t- α) as t ↓ 0 for any α > 1 / γ. This is an approximate converse of a perturbation theorem for this class of semigroups. © 2006 Elsevier Inc. All rights reserved. |
| first_indexed | 2024-03-06T21:22:45Z |
| format | Journal article |
| id | oxford-uuid:42091171-5772-4325-8233-e75851156348 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-06T21:22:45Z |
| publishDate | 2006 |
| publisher | Elsevier |
| record_format | dspace |
| spelling | oxford-uuid:42091171-5772-4325-8233-e758511563482022-03-26T14:47:10ZRank-1 perturbations of cosine functions and semigroupsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:42091171-5772-4325-8233-e75851156348EnglishSymplectic Elements at OxfordElsevier2006Batty, CArendt, WLet A be the generator of a cosine function on a Banach space X. In many cases, for example if X is a UMD-space, A + B generates a cosine function for each B ∈ L (D ((ω - A)1 / 2), X). If A is unbounded and frac(1, 2) < γ ≤ 1, then we show that there exists a rank-1 operator B ∈ L (D ((ω - A)γ), X) such that A + B does not generate a cosine function. The proof depends on a modification of a Baire argument due to Desch and Schappacher. It also allows us to prove the following. If A + B generates a distribution semigroup for each operator B ∈ L (D (A), X) of rank-1, then A generates a holomorphic C0-semigroup. If A + B generates a C0-semigroup for each operator B ∈ L (D ((ω - A)γ), X) of rank-1 where 0 < γ < 1, then the semigroup T generated by A is differentiable and {norm of matrix} T′ (t) {norm of matrix} = O (t- α) as t ↓ 0 for any α > 1 / γ. This is an approximate converse of a perturbation theorem for this class of semigroups. © 2006 Elsevier Inc. All rights reserved. |
| spellingShingle | Batty, C Arendt, W Rank-1 perturbations of cosine functions and semigroups |
| title | Rank-1 perturbations of cosine functions and semigroups |
| title_full | Rank-1 perturbations of cosine functions and semigroups |
| title_fullStr | Rank-1 perturbations of cosine functions and semigroups |
| title_full_unstemmed | Rank-1 perturbations of cosine functions and semigroups |
| title_short | Rank-1 perturbations of cosine functions and semigroups |
| title_sort | rank 1 perturbations of cosine functions and semigroups |
| work_keys_str_mv | AT battyc rank1perturbationsofcosinefunctionsandsemigroups AT arendtw rank1perturbationsofcosinefunctionsandsemigroups |