The Exponential of Quasi Block-Toeplitz Matrices

Abstract

The matrix Wiener algebra, ${{\cal W}_N}: = {{\rm{M}}_N}({\cal W})$ of order N > 0, is the matrix algebra formed by N × N matrices whose entries belong to the classical Wiener algebra ${\cal W}$ of functions with absolutely convergent Fourier series. A block-Toeplitz matrix T(a) = [A_i,j]_i,j≥0 is a block semi-infinite matrix such that its blocks A_i,j are finite matrices of order N, A_i,j = A_r,s whenever i − j = r − s and its entries are the coefficients of the Fourier expansion of the generator $a: \mathbb{T}\rightarrow {{\rm{M}}_N}(\mathbb{C})$. Such a matrix can be regarded as a bounded linear operator acting on the direct sum of N copies of ${L^2}(\mathbb{T})$. We show that exp(T(a)) difieres from T(exp(a)) only in a compact operator with a known bound on its norm. In fact, we prove a slightly more general result: for every entire function f and for every compact operator E, there exists a compact operator F such that f (T (a) + E) = T (f (a)) + F. We call these T (a) + E′s matrices, the quasi block-Toeplitz matrices, and we show that via a computation-friendly norm, they form a Banach algebra. Our results generalize and are motivated by some recent results of Dario Andrea Bini, Stefano Massei and Beatrice Meini.