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Register a new userAsymptotics of determinants for finite sections of operators with almost periodic diagonals
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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.
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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.
Let $T^n$ denote the n-dimensional torus. The class of the bounded operators on $L^2(T^n)$ with analytic orbit under the action of $T^n$ by conjugation with the translation operators is shown to coincide with the class of the zero-order pseudodifferential operators on $T^n$ whose discrete symbol $(a_j)_{jin Z^n}$ is uniformly analytic, in the sense that there exists $C>1$ such that the derivatives of $a_j$ satisfy $|partial^alpha a_j(x)|leq C^{1+|alpha|}alpha!$ for all $xin T^n$, all $jin Z^n$ and all $alphain N^n$. This implies that this class of pseudodifferential operators is a spectrally invariant *-subalgebra of the algebra of all bounded operators on $L^2(T^n)$.
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.
Lins theorem states that for all $epsilon > 0$, there is a $delta > 0$ such that for all $n geq 1$ if self-adjoint contractions $A,B in M_n(mathbb{C})$ satisfy $|[A,B]|leq delta$ then there are self-adjoint contractions $A,B in M_n(mathbb{C})$ with $[A,B]=0$ and $|A-A|,|B-B|<epsilon$. We present full details of the approach in arXiv:0808.2474, which seemingly is the closest result to a general constructive proof of Lins theorem. Constructive results for some special cases are presented along with applications to the problem of almost commuting matrices where $B$ is assumed to be normal and also to macroscopic observables.
A Banach space X has the SHAI (surjective homomorphisms are injective) property provided that for every Banach space Y, every continuous surjective algebra homomorphism from the bounded linear operators on X onto the bounded linear operators on Y is injective. The main result gives a sufficient condition for X to have the SHAI property. The condition is satisfied for L^p (0, 1) for 1 < p < infty, spaces with symmetric bases that have finite cotype, and the Schatten p-spaces for 1 < p < infty.
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