Vol. 86, No. 2, 1980

Recent Issues
Vol. 294: 1
Vol. 293: 1  2
Vol. 292: 1  2
Vol. 291: 1  2
Vol. 290: 1  2
Vol. 289: 1  2
Vol. 288: 1  2
Vol. 287: 1  2
Online Archive
The Journal
Editorial Board
Special Issues
Submission Guidelines
Submission Form
Author Index
To Appear
ISSN: 0030-8730
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