Toeplitz determinants with several Fisher-Hartwig singularities. (English) Zbl 0910.47020

Ehrhardt, Torsten

Chemnitz: TU Chemnitz, Fakultät für Mathematik der TU Chemnitz, 111 p. (1997).

Summary: A conjecture raised by M. E. Fisher and R. E. Hartwig describes the asymptotic behavior of Toeplitz determinants for a class of generating functions. This conjecture has been proved in many important cases. In other cases, however, it has turned out to fail and has been reformulated.
The main topic of the present dissertation is the proof of the Fisher-Hartwig conjecture under certain smoothness assumptions in all the cases in which it is expected to be true. The proof relies on operator theoretical methods, and its presentation given here is selfcontained. An important auxiliary result is a separation theorem, which is also of interest in its own right. Moreover, certain asymptotic properties of the inverses of finite Toeplitz matrices play a dominant role.
A classification and an explicit description of the generating functions in view of the validity of the original conjecture is also given. Furthermore, a generalization into another direction is pursued by replacing the generating functions by distributions. Finally, a result concerning the asymptotics of Toeplitz determinants is established in a trivial case where the original conjecture fails.

MSC:

47B35	Toeplitz operators, Hankel operators, Wiener-Hopf operators
47B10	Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
15A15	Determinants, permanents, traces, other special matrix functions

Keywords:

Szegő limit theorem; asymptotic behavior of finite Toeplitz matrices; Toeplitz operators; Hankel operators; operator determinants; asymptotic behavior of Toeplitz determinants; Fisher-Hartwig conjecture; smoothness assumptions; separation theorem