In this paper we consider solving Hermitian Toeplitz systems Tnx= b by using the preconditioned conjugate gradient (PCG) method. Here the Toeplitz matrices Tn are assumed to be generated by a non-negative continuous 2 -periodic function f, i.e. Tn =Tn[f]. It was proved in (Linear Algebra Appl. 1993; 190:181) that if f is positive then the spectrum of Tn[1=f]Tn[f] is clustered around 1. We prove...

A matrix A of size n is called g-circulant if A = [ a(r−gs) mod n ]n−1 r,s=0 , while a matrix A is called g-Toeplitz if its entries obey the rule A = [ar−gs] n−1 r,s=0. In this note we study the eigenvalues of g-circulants and we provide a preliminary asymptotic analysis of the eigenvalue distribution of g-Toeplitz sequences, in the case where the numbers {ak} are the Fourier coefficients of an...

The authors of the title proved in [2] an elegant identity expressing a Toeplitz determinant in terms of the Fredholm determinant of an infinite matrix which (although not described as such) is the product of two Hankel matrices. The proof used combinatorial theory, in particular a theorem of Gessel expressing a Toeplitz determinant as a sum over partitions of products of Schur functions. The p...

Abstract. We calculate the boundary correlation function of fixed-to-free boundary condition changing operators in the square-lattice Ising model. The correlation function is expressed in four different ways using 2×2 block Toeplitz determinants. We show that these can be transformed into a scalar Toeplitz determinant when the size of the matrix is even. To know the asymptotic behavior of the c...

In this paper we prove that if S equals a finite sum of finite products of Toeplitz operators on the Bergman space of the unit disk, then S is compact if and only if the Berezin transform of S equals 0 on ∂D. This result is new even when S equals a single Toeplitz operator. Our main result can be used to prove, via a unified approach, several previously known results about compact Toeplitz oper...

A family of recently discovered commutative C∗-algebras of Toeplitz operators on the unit disk can be classified as follows. Each pencil of hyperbolic straight lines determines a set of symbols consisting of functions which are constant on the corresponding cycles, the orthogonal trajectories to lines forming a pencil. The C∗-algebra generated by Toeplitz operators with such symbols turns out t...

In this talk, based on the generalized locally Toeplitz (GLT) theory, we present a framework to construct an algebraic multigrid (AMG) solver for matrices with Toeplitz structure. We recall the basics of the AMG algorithm and of the GLT theory, together with the properties of a Toeplitz matrix. Based on GLT we show how to construct an AMG solver for such matrices (GLT-MG) and discuss its main a...

For random selfadjoint (real symmetric, complex Hermitian, or quaternion self-dual) Toeplitz matrices and real symmetric Hankel matrices, the existence of universal limit distributions for eigenvalues and products of several independent matrices is proved. The joint moments are the integral sums related to certain pair partitions. Our method can apply to random Hankel and Toeplitz band matrices...

We study the structure of the C∗-algebra generated by Toeplitz operators with piece-wise continuous symbols, putting a special emphasis to Toeplitz operators with unbounded symbols. We show that none of a finite sum of finite products of the initial generators is a compact perturbation of a Toeplitz operator. At the same time the uniform closure of the set of such sum of products contains a hug...

We combine some basic techniques of linear algebra with some expressions for Toeplitz and circulant matrices and the properties of Gaussian random matrices to estimate the norms of Gaussian Toeplitz and circulant random matrices and their inverses. In the case of circulant matrices we obtain sharp probabilistic estimates, which show that the matrices are expected to be very well conditioned. Ou...