We are interested in permutations preserving certain distribution properties of sequences. In particular we consider $\mu$-uniformly distributed sequences on a compact metric space $X$, 0-1 sequences with densities, and Cesàro summable bounded sequences. It is shown that the maximal subgroups, respectively subsemigroups, of $Aut\left(\mathbf{N}\right)$ leaving any of the above spaces invariant coincide. A subgroup of these permutation groups, which can be determined explicitly, is the Lévy group $𝒢$. We show that $𝒢$ is big in the...

We distinguish a class of unbounded operators in ${}^r$, r ≥ 1, related to the self-adjoint operators in ². For these operators we prove a kind of individual ergodic theorem, replacing the classical Cesàro averages by Borel summability. The result is equivalent to a version of Gaposhkin’s criterion for the a.e. convergence of operators. In the proof, the theory of martingales and interpolation in ${}^r$-spaces are applied.

In this paper, $K$ denotes a complete, non-trivially valued, non-archimedean field. Sequences and infinite matrices have entries in $K.$ The main purpose of this paper is to prove some product theorems involving the methods $M$ and $(N,p_n)$ in such fields $K.$