Publicação
Toeplitz operators with analytic symbols and corona problems
| Resumo: | The study of the properties of a Toeplitz operator TG with 2 x 2 symbol G is related with an appropriate factorization of G. In particular, TG is Fredholm if and only if G admits a Wiener-Hopf (WH) factorization and it is invertible if and only if this factorization is canonical. For almost periodic (AP) symbols, the so-called AP factorization appears as a natural generalization of the WH factorization, which does not exist for such matrices unless it is canonical. The existence and the actual determination of those factorizations are shown to be closely related to certain corona problems whose data are particular solutions to a Riemann-Hilbert problem Gh+ = h_, h± Є (H±∞)². In this thesis, on the one hand, conditions are established which are equivalent to the corona conditions but easier to verify, if G 1 are analytic and bounded in a strip of the complex plane. On the other and, new classes of symbols G, for which a non-trivial solution to the Riemann-Hilbert problem can be explicitly determined and the corona conditions can be veri ed by the above mentioned approach, are identi ed. Criteria for factorability of matrix function G in those classes are thus obtained. |
|---|---|
| Autores principais: | Diogo, Cristina Isabel Correia |
| Assunto: | Toeplitz operator Riemann-Hilbert problem corona theorem Wiener-Hopf factorization AP factorization Operador de Toeplitz Problema de Riemann-Hilbert Teorema da coroa Factorização de Wiener-Hopf Factorização AP |
| Ano: | 2009 |
| País: | Portugal |
| Tipo de documento: | tese de doutoramento |
| Tipo de acesso: | acesso aberto |
| Instituição associada: | Universidade do Minho |
| Idioma: | inglês |
| Origem: | RepositóriUM - Universidade do Minho |
| Resumo: | The study of the properties of a Toeplitz operator TG with 2 x 2 symbol G is related with an appropriate factorization of G. In particular, TG is Fredholm if and only if G admits a Wiener-Hopf (WH) factorization and it is invertible if and only if this factorization is canonical. For almost periodic (AP) symbols, the so-called AP factorization appears as a natural generalization of the WH factorization, which does not exist for such matrices unless it is canonical. The existence and the actual determination of those factorizations are shown to be closely related to certain corona problems whose data are particular solutions to a Riemann-Hilbert problem Gh+ = h_, h± Є (H±∞)². In this thesis, on the one hand, conditions are established which are equivalent to the corona conditions but easier to verify, if G 1 are analytic and bounded in a strip of the complex plane. On the other and, new classes of symbols G, for which a non-trivial solution to the Riemann-Hilbert problem can be explicitly determined and the corona conditions can be veri ed by the above mentioned approach, are identi ed. Criteria for factorability of matrix function G in those classes are thus obtained. |
|---|