Aspects of Toeplitz operators and matrices : asymptotics, norms, singular values
The research contained in this thesis can be divided into two related, but distinct parts. The rst chapter deals with block Toeplitz operators de ned by rational matrix function symbols on discrete sequence spaces. Here we study sequences of operators that converge to the inverses of these Toeplitz operators via an invertibility result involving a special representation of the symbol of these block Toeplitz operators. The second part focuses on a special class of matrices generated by banded Toeplitz matrices, i.e., Toeplitz matrices with a nite amount of non-zero diagonals. The spectral theory of banded Toeplitz matrices is well developed, and applied to solve questions regarding the behaviour of the singular values of Toeplitz-generated matrices. In particular, we use the behaviour of the singular values to deduce bounds for the growth of the norm of the inverse of Toeplitz-generated matrices. In chapter 2, we use a special state-space representation of a rational matrix function on the unit circle to de ne a block Toeplitz operator on a discrete sequence space. A discrete Riccati equation can be associated with this representation which can be used to prove an invertibility theorem for these Toeplitz operators. Explicit formulas for the inverse of the Toeplitz operators are also derived that we use to de ne a sequence of operators that converge in norm to the inverse of the Toeplitz operator. The rate of this convergence, as well as that of a related Riccati di erence equation is also studied. We conclude with an algorithm for the inversion of the nite sections of block Toeplitz operators. Chapter 3 contains the main research contribution of this thesis. Here we derive sharp growth rates for the norms of the inverses of Toeplitz-generated matrices. These results are achieved by employing powerful theory related to the Avram-Parter theorem that describes the distribution of the singular values of banded Toeplitz matrices. The investigation is then extended to include the behaviour of the extreme and general singular values of Toeplitz-generated matrices. We conclude with Chapter 4, which sets out to answer a very speci c question regarding the singular vectors of a particular subclass of Toeplitz-generated matrices. The entries of each singular vector seems to be a permutation (up to sign) of the same set of real numbers. To arrive at an explanation for this phenomenon, explicit formulas are derived for the singular values of the banded Toeplitz matrices that serve as generators for the matrices in question. Some abstract algebra is also employed together with some results from the previous chapter to describe the permutation phenomenon. Explicit formulas are also shown to exist for the inverses of these particular Toeplitz-generated matrices as well as algorithms to calculate the norms and norms of the inverses. Finally, some additional results are compiled in an appendix.