Vol. 86, No. 2, 1980

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One-parameter semigroups of isometries into Hp

Earl Robert Berkson

Vol. 86 (1980), No. 2, 403–413

In this paper we explicitly describe all strongly continuous one-parameter semigroups {Tt} of isometries of Hp(D) into Hp(D), where 1 p < , p2, 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.

Mathematical Subject Classification 2000
Primary: 47D05, 47D05
Secondary: 30D55, 46J15, 47B37
Received: 3 January 1979
Published: 1 February 1980
Earl Robert Berkson