Rates of decay in the classical Katznelson-Tzafriri theorem
The Katznelson-Tzafriri Theorem states that, given a powerbounded operator T , T n(I − T ) → 0 as n → ∞ if and only if the spectrum σ(T ) of T intersects the unit circle T in at most the point 1. This paper investigates the rate at which decay takes place when σ(T ) ∩ T = {1}. The results obtained l...
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| Materialtyp: | Journal article |
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Hebrew University Magnes Press
2016
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| _version_ | 1826304966925484032 |
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| author | Seifert, D |
| author_facet | Seifert, D |
| author_sort | Seifert, D |
| collection | OXFORD |
| description | The Katznelson-Tzafriri Theorem states that, given a powerbounded operator T , T n(I − T ) → 0 as n → ∞ if and only if the spectrum σ(T ) of T intersects the unit circle T in at most the point 1. This paper investigates the rate at which decay takes place when σ(T ) ∩ T = {1}. The results obtained lead, in particular, to both upper and lower bounds on this rate of decay in terms of the growth of the resolvent operator R(eiθ , T ) as θ → 0. In the special case of polynomial resolvent growth, these bounds are then shown to be optimal for general Banach spaces but not in the Hilbert space case. |
| first_indexed | 2024-03-07T06:25:44Z |
| format | Journal article |
| id | oxford-uuid:f43bdb20-4e15-42c3-b3b9-bcc35a34c512 |
| institution | University of Oxford |
| last_indexed | 2024-03-07T06:25:44Z |
| publishDate | 2016 |
| publisher | Hebrew University Magnes Press |
| record_format | dspace |
| spelling | oxford-uuid:f43bdb20-4e15-42c3-b3b9-bcc35a34c5122022-03-27T12:18:08ZRates of decay in the classical Katznelson-Tzafriri theoremJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:f43bdb20-4e15-42c3-b3b9-bcc35a34c512Symplectic Elements at OxfordHebrew University Magnes Press2016Seifert, DThe Katznelson-Tzafriri Theorem states that, given a powerbounded operator T , T n(I − T ) → 0 as n → ∞ if and only if the spectrum σ(T ) of T intersects the unit circle T in at most the point 1. This paper investigates the rate at which decay takes place when σ(T ) ∩ T = {1}. The results obtained lead, in particular, to both upper and lower bounds on this rate of decay in terms of the growth of the resolvent operator R(eiθ , T ) as θ → 0. In the special case of polynomial resolvent growth, these bounds are then shown to be optimal for general Banach spaces but not in the Hilbert space case. |
| spellingShingle | Seifert, D Rates of decay in the classical Katznelson-Tzafriri theorem |
| title | Rates of decay in the classical Katznelson-Tzafriri theorem |
| title_full | Rates of decay in the classical Katznelson-Tzafriri theorem |
| title_fullStr | Rates of decay in the classical Katznelson-Tzafriri theorem |
| title_full_unstemmed | Rates of decay in the classical Katznelson-Tzafriri theorem |
| title_short | Rates of decay in the classical Katznelson-Tzafriri theorem |
| title_sort | rates of decay in the classical katznelson tzafriri theorem |
| work_keys_str_mv | AT seifertd ratesofdecayintheclassicalkatznelsontzafriritheorem |