Arrangements of series preserving their convergence or boundedness

For a map $\rho$ of $\mathbb{N}$ into itself, consider the induced transformation $\sum _n{x}_n\mapsto \sum _n{x}_{{\rho }_{(}n)}$ of series in a topological vector space. Then such properties of this transformation as sending convergent series to convergent series, or convergent series to bounded series, or bounded series to bounded series (and a few more) are mutually equivalent. Moreover, they are equivalent to an intrinsic property of ρ which reduces to those found by Agnew and Pleasants (in the case of permutations) and Wituła (in the general case) as necessary and sufficient conditions for the above transformation to preserve convergence of scalar series. In the paper, the scalar case is treated first using simple Banach space methods, and then the result is easily extended to the general setting.