Showing papers by "Albrecht Böttcher published in 2012"

Eigenvalues of Hermitian Toeplitz matrices with polynomially increasing entries

Abstract: The paper is concerned with finite Hermitian Toeplitz matrices whose entries in the first row grow like a polynomial. Such matrices cannot be viewed as truncations of an infinite Toeplitz matrix which is generated by an integrable function or a nice measure. The main results describe the first-order asymptotics of the extreme eigenvalues as the matrix dimension goes to infinity and also deliver unexpected barriers for the eigenvalues. One purpose of the paper is to popularize once more that questions on the eigenvalues of matrices can be answered in a very elegant way by passing to integral operators. This idea was introduced by Harold Widom about fifty years ago. In this way one can also give an alternative proof to results by William F. Trench on Hermitian Toeplitz matrices with increasing entries.

Asymptotics of Individual Eigenvalues of a Class of Large Hessenberg Toeplitz Matrices

Abstract: We study the asymptotic behavior of individual eigenvalues of the n-by-n truncations of certain infinite Hessenberg Toeplitz matrices as n goes to infinity.

Orthogonal and Skew-Symmetric Operators in Real Hilbert Space

Abstract: In the theory of traces on operator ideals, it is desirable to treat not only the complex case. Several proofs become much easier when the underlying operators are represented by real matrices. Motivated by this observation, we prove two theorems which, to the best of our knowledge, are not available in the real setting: (1) every operator is a finite linear combination of orthogonal operators, and (2) every skew-symmetric compact operator S is a commutator [A, T], where certain properties of S are inherited to T. In our opinion, theses results are interesting for their own sake. They will also be used in future studies of trace theory by the second-named author.

First-order Trace Formulae for the Iterates of the Fox–Li Operator

Abstract: The paper is devoted to first-order trace formulas for the iterates of the Fox–Li and related Wiener–Hopf integral operators. Such formulas provide first insight into the asymptotic behaviour of the eigenvalues and can be used to test whether a specific guess for the eigenvalue distribution is acceptable or not. The main technical problem consists in obtaining the asymptotics of a multivariate oscillatory integral whose stationary points constitute a line.

The Fox-Li operator as a test and a spur for Wiener-Hopf theory

Abstract: The paper is a concise survey of some rigorous results on the Fox–Li operator. This operator may be interpreted as a large truncation of a Wiener–Hopf operator with an oscillating symbol. Employing theorems from Wiener–Hopf theory one can therefore derive remarkable properties of the Fox–Li operator in a fairly comfortable way, but it turns out that Wiener–Hopf theory is unequal to the task of answering the crucial questions on the Fox–Li operator.