In this paper we are interested in generalizing Keller-type eigenvalue estimates for the non-selfadjoint Schrödinger operator to Dirac operator, imposing some suitable rigidity conditions on matricial structure of potential, without necessarily requiring smallness its norm.

We present a scenario concerning the existence of a large class of reflectionless selfadjoint analytic difference operators. In order to exemplify this scenario, we summarize our results on reflectionless self-adjoint difference operators of relativistic CalogeroMoser type.

We show that the property T factors associated with groups SL(n)(Z), n > or = 3 are generated by two selfadjoint elements, one of which has an arbitrarily small support. This answers a question of Dan Voiculescu.