Let{T(t)}t≥0beaC0-semigrouponaseparableHilbertspaceH.Weshow thatT(t) is anm-isometry for any t if andonly if themapping t∈R+→ T(t)x 2 foreachx∈Hisapolynomialofdegreeatmostm.Thispropertyis usedtostudym-isometricrighttranslationsemigrouponweightedLp-spaces. Wealsoprovidealternativecharacterizationsof theabovepropertybyimposingconditionsonthe infinitesimalgeneratoroperatorandonthecogenerator operatorof{T(t)}t≥0.Moreover,weprovethatanon-unitary2-isometryTon aHilbertspacesatisfyingthekernelcondition, that is, T∗T(KerT∗)⊂KerT∗ , canbeembeddedintoaC0-semigroupifandonlyifdim(KerT∗)=∞.
