We study free profinite subgroups of free profinite semigroups of the same rank using, as main tools, iterated implicit operators, subword complexity and the associated entropy.

Yosida's definition of potential operators for semigroups [17] makes it possible to deal with transient Markov processes and a class of recurrent Markov processes in a unified operator theoretical way. In this paper, we prove some general properties of his potential operators, show which Markov processes admit the potential operators, and investigate the cases of processes with stationary indep...

The notion of automatic selfadjointness all ideals in a multiplicative semigroup the bounded linear operators on separable Hilbert space B(H) arose 2015 discussion with Heydar Radjavi who pointed out that and finite rank F(H) possessed this unitary invariant property which category we named SI semigroups (for selfadjoint ideal semigroups). Equivalent to is solvability, for each A semigroup, bil...

We construct  a noncommutative analog of additive functional equations on discrete quantum semigroups and show that this noncommutative functional equation has Hyers-Ulam stability on amenable discrete quantum semigroups. The discrete quantum semigroups that we consider in this paper are in the sense of van Daele, and the amenability is in the sense of Bèdos-Murphy-Tuset. Our main result genera...

For semigroups and for bounded operators we introduce the new notion of Bergman distance. Systems with a finite Bergman distance share the same stability properties, and the Bergman distance is preserved under the Cayley transform. This way, we get stability results in continuous and discrete time. As an example, we show that bounded perturbations lead to pairs of semigroups with finite Bergman...