dc.contributor.advisor Ter Horst, S. dc.contributor.advisor Ran, A.C.M. dc.contributor.advisor Groenewald, G.J. dc.contributor.author Jaftha, Jacob Jacobus dc.date.accessioned 2020-07-16T14:07:52Z dc.date.available 2020-07-16T14:07:52Z dc.date.issued 2020 dc.identifier.uri https://orcid.org/0000-0001-9643-6113 dc.identifier.uri http://hdl.handle.net/10394/35161 dc.description PhD (Mathematics), North-West University, Potchefstroom Campus en_US dc.description.abstract Let Hp be the Hardy space of p-integrable functions on the unit circle T in the complex plane that have an analytic extension to the open unit disk D. Suppose that ω is a rational function with poles on the unit circle. The topic of this thesis is the analysis of a Toeplitz-like operator Tω in Hp generated by such an ω. We investigate Fredholm properties, the spectrum and the adjoint in case ω is a scalar function and explore the Fredholm properties of TΩ in case Ω is a rational matrix function with poles on T. We show that, in general, the operator Tω is a well-defined, closed, densely defined linear Operator whose domain contains the polynomials. It is shown that the operator is Fredholm if and only if the symbol has no zeroes on the unit circle, and a formula for the index is given as well. A matrix representation of the operator is discussed. en_US A description of the spectrum of Tω and its various parts, i.e., point, residual and conTenuous spectrum, is given, as well as a description of the essential spectrum. In this case, it is shown that the essential spectrum need not to be connected in C. Various examples illustrate the results. The adjoint operator T∗ω is described. In the case where p = 2and ω has poles only on the unit circle T, a description is given for when T∗ ω is symmetric and when T∗ ω admits a self adjoint extension. We compare the operator with unbounded Toeplitz operators studied earlier and show that if ω is a properration alfunction, then T∗ ω coincides with an unbounded Toeplitz operator studied earlier by Sarason. We extend the analysis of the Toeplitz-like operator to the case where it is generated By a rational matrix function having poles on T. A Wiener-Hopf type factorization of rational matrix functions with poles and zeroes on T is introduced and then used to analyse the FredholmpropertiesofToeplitz-likeoperators.Aformulafortheindex,basedonthe factorization, is given. Furthermore, it is shown that the determinant of ω having no zeroes on T is not sufficient for Tω being Fredholm, which is in contrast to the classical case, where the symbol has no zeroes on T is sufficient for the operator Tω being Fredholm. dc.language.iso en en_US dc.publisher North-West University (South Africa) en_US dc.subject Toeplitzoperators en_US dc.subject Unboundedoperators en_US dc.subject HardySpace en_US dc.subject Fredholm operators en_US dc.subject Adjoint operator en_US dc.subject Wiener-Hopffactorization en_US dc.subject Rationalsymbol en_US dc.subject Toeplitzoperators en_US dc.subject Unbounded operators en_US dc.subject Hardy Space en_US dc.subject Fredholm operators en_US dc.subject Adjoint operator en_US dc.subject Wiener-Hopffactorization en_US dc.subject Rational symbol en_US dc.title Toeplitz-like operators with rational symbol having poles on the unit circle en_US dc.type Thesis en_US dc.description.thesistype Doctoral en_US dc.contributor.researchID 24116327 - Ter Horst, Sanne (Supervisor) dc.contributor.researchID 20000212 - Ran, Andreas Cornelis Maria (Supervisor) dc.contributor.researchID 12066680 - Groenewald, Gilbert Joseph (Supervisor)
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