On the exponential of semi-infinite quasi-Toeplitz matrices

Let a(z) = ∑ i ∈ Zaizi be a complex valued function defined for | z| = 1 , such that ∑ i ∈ Z| ai| < ∞; define T(a)=(ti,j)i,j∈Z+,ti,j=aj-i for i, j∈ Z+, the semi-infinite Toeplitz matrix associated with the symbol a(z); let E=(ei,j)i,j∈Z+ be a compact operator in ℓp, with 1 ≤ p≤ ∞. A semi-infinite matrix of the kind A= T(a) + E is said quasi-Toeplitz (QT). The problem of the computation of exp (A) or exp (A) v, with A quasi-Toeplitz and v a vector, arises in many applications. We prove that the exponential of a QT-matrix A is QT, that is, exp (A) = T(exp (a)) + F where F is a compact operator in ℓp. This property allows the design of an algorithm for computing exp (A) and exp (A) v up to any precision. The case of families of n× n matrices obtained by truncating infinite QT-matrices to finite size is also considered. Numerical experiments show the effectiveness of this approach. © 2018, Springer-Verlag GmbH Germany, part of Springer Nature.