Operators on banach spaces of Bourgain-Delbaen type
<p>The research in this thesis was initially motivated by an outstanding problem posed by Argyros and Haydon. They used a generalised version of the Bourgain-Delbaen construction to construct a Banach space $XK$ for which the only bounded linear operators on $XK$ are compact perturbations of (...
| المؤلف الرئيسي: | |
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| مؤلفون آخرون: | |
| التنسيق: | أطروحة |
| اللغة: | English |
| منشور في: |
2013
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| الموضوعات: |
| _version_ | 1826316598609182720 |
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| author | Tarbard, M |
| author2 | Haydon, R |
| author_facet | Haydon, R Tarbard, M |
| author_sort | Tarbard, M |
| collection | OXFORD |
| description | <p>The research in this thesis was initially motivated by an outstanding problem posed by Argyros and Haydon. They used a generalised version of the Bourgain-Delbaen construction to construct a Banach space $XK$ for which the only bounded linear operators on $XK$ are compact perturbations of (scalar multiples of) the identity; we say that a space with this property has very few operators. The space $XK$ possesses a number of additional interesting properties, most notably, it has $ell_1$ dual. Since $ell_1$ possesses the Schur property, weakly compact and norm compact operators on $XK$ coincide. Combined with the other properties of the Argyros-Haydon space, it is tempting to conjecture that such a space must necessarily have very few operators. Curiously however, the proof that $XK$ has very few operators made no use of the Schur property of $ell_1$. We therefore arrive at the following question (originally posed in cite{AH}): must a HI, $mathcal{L}_{infty}$, $ell_1$ predual with few operators (every operator is a strictly singular perturbation of $lambda I$) necessarily have very few operators?</p> <p>We begin by giving a detailed exposition of the original Bourgain-Delbaen construction and the generalised construction due to Argyros and Haydon. We show how these two constructions are related, and as a corollary, are able to prove that there exists some $delta > 0$ and an uncountable set of isometries on the original Bourgain-Delbaen spaces which are pairwise distance $delta$ apart.</p> <p>We subsequently extend these ideas to obtain our main results. We construct new Banach spaces of Bourgain-Delbaen type, all of which have $ell_1$ dual. The first class of spaces are HI and possess few, but not very few operators. We thus have a negative solution to the Argyros-Haydon question. We remark that all these spaces have finite dimensional Calkin algebra, and we investigate the corollaries of this result. We also construct a space with $ell_1$ Calkin algebra and show that whilst this space is still of Bourgain-Delbaen type with $ell_1$ dual, it behaves somewhat differently to the first class of spaces.</p> <p>Finally, we briefly consider shift-invariant $ell_1$ preduals, and hint at how one might use the Bourgain-Delbaen construction to produce new, exotic examples.</p> |
| first_indexed | 2024-03-07T07:42:17Z |
| format | Thesis |
| id | oxford-uuid:4be220be-9347-48a1-85e6-eb0a30a8d51a |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-12-09T03:47:55Z |
| publishDate | 2013 |
| record_format | dspace |
| spelling | oxford-uuid:4be220be-9347-48a1-85e6-eb0a30a8d51a2024-12-08T10:45:03ZOperators on banach spaces of Bourgain-Delbaen typeThesishttp://purl.org/coar/resource_type/c_db06uuid:4be220be-9347-48a1-85e6-eb0a30a8d51aFunctional analysis (mathematics)EnglishOxford University Research Archive - Valet2013Tarbard, MHaydon, R<p>The research in this thesis was initially motivated by an outstanding problem posed by Argyros and Haydon. They used a generalised version of the Bourgain-Delbaen construction to construct a Banach space $XK$ for which the only bounded linear operators on $XK$ are compact perturbations of (scalar multiples of) the identity; we say that a space with this property has very few operators. The space $XK$ possesses a number of additional interesting properties, most notably, it has $ell_1$ dual. Since $ell_1$ possesses the Schur property, weakly compact and norm compact operators on $XK$ coincide. Combined with the other properties of the Argyros-Haydon space, it is tempting to conjecture that such a space must necessarily have very few operators. Curiously however, the proof that $XK$ has very few operators made no use of the Schur property of $ell_1$. We therefore arrive at the following question (originally posed in cite{AH}): must a HI, $mathcal{L}_{infty}$, $ell_1$ predual with few operators (every operator is a strictly singular perturbation of $lambda I$) necessarily have very few operators?</p> <p>We begin by giving a detailed exposition of the original Bourgain-Delbaen construction and the generalised construction due to Argyros and Haydon. We show how these two constructions are related, and as a corollary, are able to prove that there exists some $delta > 0$ and an uncountable set of isometries on the original Bourgain-Delbaen spaces which are pairwise distance $delta$ apart.</p> <p>We subsequently extend these ideas to obtain our main results. We construct new Banach spaces of Bourgain-Delbaen type, all of which have $ell_1$ dual. The first class of spaces are HI and possess few, but not very few operators. We thus have a negative solution to the Argyros-Haydon question. We remark that all these spaces have finite dimensional Calkin algebra, and we investigate the corollaries of this result. We also construct a space with $ell_1$ Calkin algebra and show that whilst this space is still of Bourgain-Delbaen type with $ell_1$ dual, it behaves somewhat differently to the first class of spaces.</p> <p>Finally, we briefly consider shift-invariant $ell_1$ preduals, and hint at how one might use the Bourgain-Delbaen construction to produce new, exotic examples.</p> |
| spellingShingle | Functional analysis (mathematics) Tarbard, M Operators on banach spaces of Bourgain-Delbaen type |
| title | Operators on banach spaces of Bourgain-Delbaen type |
| title_full | Operators on banach spaces of Bourgain-Delbaen type |
| title_fullStr | Operators on banach spaces of Bourgain-Delbaen type |
| title_full_unstemmed | Operators on banach spaces of Bourgain-Delbaen type |
| title_short | Operators on banach spaces of Bourgain-Delbaen type |
| title_sort | operators on banach spaces of bourgain delbaen type |
| topic | Functional analysis (mathematics) |
| work_keys_str_mv | AT tarbardm operatorsonbanachspacesofbourgaindelbaentype |