In this paper we explicitly
describe all strongly continuous one-parameter semigroups {Tt} of isometries of
Hp(D) into Hp(D), where 1 ≦ p < ∞, p≠2, and D is the unit disc |z| < 1 in the
complex plane C. It turns out (Theorem (1.6)) that for each t, Tt= ψtUt, where Ut
is a surjective isometry and ψt is an inner function (the families {ψt} and {Ut} are
uniquely determined provided {Ut} is suitably normalized). The nature of the
family {ψt} depends on the set of common fixed points of the family of
Möbius transformations of the disc associated with the family {Ut}. If
there is exactly one common fixed point in D, then {Tt} must consist of
surjective isometries (§4); otherwise {Tt} consists of surjective isometries only in
very special cases (§§2,5). The families {ψt} are explicitly described in this
paper.
