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, A N ( a ) , with a being in a Wiener type algebra and defined on an annular cylinder ( [ 0 , 1 ] 2 × T ) , widely generalize the sequences of finite sections of a Toeplitz operator. We prove that if a ( x , x , t ) does not vanish for every ( x , t ) ∈ [ 0 , 1 ] × T then the singular values of A N ( 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 A N ( a ) converge to zero as 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 ) ) , 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. [ABSTRACT FROM AUTHOR]