On functions of quasi-toeplitz matrices

Let a(z) = ∑ _i∈ℤ ^aizi be a complex-valued function, defined for |z| = 1, such that ∑ _i=-∞^+∞ |ia_i| < ∞. Consider the semi-infinite Toeplitz matrix T(a) = (t_i,j)_i,j∈Z+ associated with the symbol a(z) such that t_i,j = a_j-i. A quasi-Toeplitz matrix associated with the symbol a(z) is a matrix of the form A = T(a)+E where E = (e_i,j), ∑ _i,j∈Z+ |e_i,j | < ∞, and is called a QT-matrix. Given a function f(x) and a QT-matrix M, we provide conditions under which f(M) is well defined and is a QT-matrix. Moreover, we introduce a parametrization of QT-matrices and algorithms for the computation of f(M). We treat the case where f(x) is given in terms of power series and the case where f(x) is defined in terms of a Cauchy integral. This analysis is also applied to finite matrices which can be written as the sum of a Toeplitz matrix and a low rank correction. Bibliography: 27 titles.