Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userThe tame symbol and determinants of Toeplitz operators
214
0
0.0
(
0
)
Authors
Efton Park
Ask ChatGPT about the research
No Arabic abstract
Suppose that $phi$ and $psi$ are smooth complex-valued functions on the circle that are invertible, have winding number zero with respect to the origin, and have meromorphic extensions to an open neighborhood of the closed unit disk. Let $T_phi$ and $T_psi$ denote the Toeplitz operators with symbols $phi$ and $psi$ respectively. We give an explicit formula for the determinant of $T_phi T_psi T_phi^{-1} T_psi^{-1}$ in terms of the products of the tame symbols of $phi$ and $psi$ on the open unit disk.
rate research
Read More
Let $A = (a_{j,k})_{j,k=-infty}^infty$ be a bounded linear operator on $l^2(mathbb{Z})$ whose diagonals $D_n(A) = (a_{j,j-n})_{j=-infty}^inftyin l^infty(mathbb{Z})$ are almost periodic sequences. For certain classes of such operators and under certain conditions, we are going to determine the asymptotics of the determinants $det A_{n_1,n_2}$ of the finite sections of the operator $A$ as their size $n_2 - n_1$ tends to infinity. Examples of such operators include block Toeplitz operators and the almost Mathieu operator.
We study the asymptotics in n for n-dimensional Toeplitz determinants whose symbols possess Fisher-Hartwig singularities on a smooth background. We prove the general non-degenerate asymptotic behavior as conjectured by Basor and Tracy. We also obtain asymptotics of Hankel determinants on a finite interval as well as determinants of Toeplitz+Hankel type. Our analysis is based on a study of the related system of orthogonal polynomials on the unit circle using the Riemann-Hilbert approach.
We obtain asymptotics for Toeplitz, Hankel, and Toeplitz+Hankel determinants whose symbols possess Fisher-Hartwig singularities. Details of the proofs will be presented in another publication.
We review some history and some recent results concerning Toeplitz determinants and their applications. We discuss, in particular, the crucial role of the two-dimensional Ising model in stimulating the development of the theory of Toeplitz determinants.
Building on techniques developed by Cowen and Gallardo-Guti{e}rrez, we find a concrete formula for the adjoint of a composition operator with rational symbol acting on the Hardy space $H^{2}$. We consider some specific examples, comparing our formula with several results that were previously known.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
