Torsten Ehrhardt

A Status Report on the Asymptotic Behavior of Toeplitz Determinants with Fisher-Hartwig Singularities

The Fisher-Hartwig conjecture describes the asymptotic behavior of Toeplitz determinants for a certain class of singular generating functions. It has been proved in many cases and reformulated in others. Recently, the author proved the conjecture in all the cases in which it can be expected to be true. In the present paper we want to give an account of the latest developments in connection with this conjecture and describe the main ideas of the proof.

Invertibility theory for Toeplitz plus Hankel operators and singular integral operators with flip

Abstract It is well known that a Toeplitz operator is invertible if and only if its symbols admits a canonical Wiener–Hopf factorization, where the factors satisfy certain conditions. A similar result holds also for singular integral operators. More generally, the dimension of the kernel and cokernel of Toeplitz or singular integral operators which and Fredholm operators can be expressed in terms of the partial indices ϰ 1 ,…,ϰ N ∈ Z of an associated Wiener–Hopf factorization problem. In this paper we establish corresponding results for Toeplitz plus Hankel operators and singular integral operators with flip under the assumption that the generating functions are sufficiently smooth (e.g., Holder continuous). We are led to a slightly different factorization problem, in which pairs (ϱ 1 ,ϰ 1 ),…,(ϱ N ,ϰ N )∈{−1,1}× Z , instead of the partial indices appear. These pairs provide the relevant information about the dimension of the kernel and cokernel and thus answer the invertibility problem.

Transformation techniques towards the factorization of non-rational 2×2 matrix functions

Abstract For the Wiener–Hopf factorization of 2×2 matrix functions G defined on a closed Carleson curve Γ , transformations G ↦ UGV where U and V are invertible rational 2×2 matrix functions are important. In the first part of this paper we establish a classification scheme for 2×2 matrix functions, which is based on such transformations. We determine invariants under these transformations and describe those matrix functions which can be transformed to triangular or Daniele–Khrapkov form. In the second part we consider special rational transformations and study the same problem. For instance, we consider transformations where U and V are rational matrix functions that are analytic and invertible on an open neighborhood of Γ . In the more complicated, but for factorization theory important case where U and V are rational matrix functions that are analytic and invertible on an open neighborhood of the closure of the domain inside of Γ or outside of Γ , respectively, the answer is slightly different.

Zero sums of idempotents in Banach algebras

The problem treated in this paper is the following.Let p1,...,pkbe idempotents in a Banach algebra B, and assume p1+...+pk=0.Does it follow that pj=0,j=1,..., k? For important classes of Banach algebras the answer turns out to be positive; in general, however, it is negative. A counterexample is given involving five nonzero bounded projections on infinite-dimensional separable Hilbert space. The number five is critical here: in Banach algebras nontrivial zero sums of four idempotents are impossible. In a purely algebraic context (no norm), the situation is different. There the critical number is four.

Logarithmic residues in Banach algebras

Letf be an analytic Banach algebra valued function and suppose that the contour integral of the logarithmic derivativef′f−1 around a Cauchy domainD vanishes. Does it follow thatf takes invertible values on all ofD? For important classes of Banach algebras, the answer is positive. In general, however, it is negative. The counterexample showing this involves a (nontrivial) zero sum of logarithmic residues (that are in fact idempotents). The analysis of such zero sums leads to results about the convex cone generated by the logarithmic residues.

Asymptotic Formulas for Determinants of a Sum of Finite Toeplitz and Hankel Matrices

The purpose of this paper is to describe asymptotic formulas for determinants of a sum of finite Toeplitz and Hankel matrices with singular generating functions. The formulas are similar to those of the analogous problem for finite Toeplitz matrices for a certain class of symbols. However, the appearance of the Hankel matrices changes the nature of the asymptotics in some instances depending on the location of the singularities. Several concrete examples are also described in the paper.

Logarithmic residues, generalized idempotents, and sums of idempotents in Banach algebras

In a commutative Banach algebraB the set of logarithmic residues (i.e., the elements that can be written as a contour integral of the logarithmic derivative of an analyticB-valued function), the set of generalized idempotents (i.e., the elements that are annihilated by a polynomial with non-negative integer simple zeros), and the set of sums of idempotents are all the same. Also, these (coinciding) sets consist of isolated points only and are closed under the operations of addition and multiplication. Counterexamples show that the commutativity condition onB is essential. The results extend to logarithmic residues of meromorphicB-valued functions.

Some identities for determinants of structured matrices

Abstract In this paper we establish several relations between the determinants of the following structured matrices: Hankel matrices, symmetric Toeplitz + Hankel matrices and Toeplitz matrices. Using known results for the asymptotic behavior of Toeplitz determinants, these identities are used in order to obtain Fisher–Hartwig type results on the asymptotics of certain skew-symmetric Toeplitz determinants and certain Hankel determinants.

Sums of Idempotents and Logarithmic Residues in Matrix Algebras

Logarithmic residues are contour integrals of logarithmic derivatives of vector-valued analytic functions. In several matrix algebras, the set of logarithmic residues coincides with the set of sums of idempotents. The connected components of this set are identified.

Asymptotic Formulas for the Determinants of Symmetric Toeplitz plus Hankel Matrices

We establish asymptotic formulas for the determinants of N × N Toeplitz plus Hankel matrices TN(o)+HN (o) as N goes to infinity for singular generating functions o defined on the unit circle in the special case where o is even, i.e., where the Toeplitz plus Hankel matrices are symmetric.