Asymptotics for Toeplitz determinants: Perturbation of symbols with a gap

Relative Szegő asymptotics for Toeplitz determinants

In this paper we study the asymptotic behavior, as n → ∞, of ratios Dn(e hdμ)/Dn(dμ) of Toeplitz determinants defined by a measure μ and a sufficiently smooth function h. The approach we follow is based on the Verblunsky coefficients associated to μ. We prove that that the second order asymptotics depends on only few Verblunsky coefficients. In particular, we establish a relative version of the...

Toeplitz operators with special symbols

Toeplitz Operators and Toeplitz Algebra with Symbols of Vanishing Oscillation

We study the C∗-algebra generated by Teoplitz operators with symbols of vanishing (mean) oscillation on the Bergman space of the unit ball. We show that the index calculation for Fredholm operators in this C∗-algebra can be easily and completely reduced to the classic case of Toeplitz operators with symbols that are continuous on the closed unit ball. Moreover, in addition to a number of other ...

Hyponormality of Toeplitz Operators with Rational Symbols

In this article we introduce a notion of ‘division’ for rational functions and then give a criterion for hyponormality of Tg+f (f, g are rational functions) in the cases where g divides f . Furthermore we show that we may assume, without loss of generality, that g divides f when we consider the hyponormality of Tg+f .

Equivariant asymptotics for Toeplitz operators

In recent years, the Tian-Zelditch asymptotic expansion for the equivariant components of the Szegö kernel of a polarized complex projective manifold, and its subsequent generalizations in terms of scaling limits, have played an important role in algebraic, symplectic, and differential geometry. A natural question is whether there exist generalizations in which the projector onto the spaces of ...