  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 semigroup defined by the
Jordan block is contractive. This yields new extensions of the classical
Fan-Taussky-Todd inequalities. As an application, we give an estimate for the
partial sums of a Bessel function.
