The Fan-Taussky-Todd inequalities and the Lumer-Phillips theorem

Abstract

We argue that a classical inequality due to Fan, Taussky and Todd (1955) is equivalent to the dissipativity of a Jordan block. As the latter can be characterised via the zeros of Chebyshev polynomials, we obtain a short new proof of the inequality. Three other inequalities of Fan–Taussky–Todd are reproven similarly. By the Lumer–Phillips theorem, the matrix semigroup generated by the Jordan block is contractive. This yields new extensions of the classical Fan–Taussky–Todd inequalities. As applications, we give an estimate for the partial sums of a Bessel function, and a contribution to the classification of self-similar Gaussian Markov processes.