  We revisit Sz.-Nagy's criteria for similarity of Hilbert space bounded linear
operators to isometries or unitaries and present new ones. We also discuss
counterparts of the Dixmier-Day theorem concerning bounded representations of
amenable groups and semigroups. We highlight the role of Folner sets in
similarity problems in both settings of unimodular, $\sigma$-compact, amenable
groups and in discrete semigroups possessing the Strong Folner condition (SFC).
