Sequences of variable-coefficient Toeplitz matrices and their singular values

The purpose of this paper is to describe asymptotic spectral properties of sequences of variable-coefficient Toeplitz matrices. These sequences, AN(a)AN(a), with a   being in a Wiener type algebra and defined on an annular cylinder ([0,1]2×T)([0,1]2×T), widely generalize the sequences of finite sections of a Toeplitz operator. We prove that if a(x,x,t)a(x,x,t) does not vanish for every (x,t)∈[0,1]×T(x,t)∈[0,1]×T then the singular values of AN(a)AN(a) have the k-splitting property, which means that, there exists an integer k such that, for N large enough, the first k  -singular values of AN(a)AN(a) converge to zero as N→∞N→∞, while the others are bounded away from zero, with k=dim⁡ker⁡T(a(0,0,t))+dim⁡ker⁡T(a(1,1,t−1))k=dim⁡ker⁡T(a(0,0,t))+dim⁡ker⁡T(a(1,1,t−1)), the sum of the kernel dimensions of two Toeplitz operators. In the end of the paper we discuss Fredholm properties of the mentioned sequences and describe them completely.