Möbius transformations of the disc and one-parameter groups of isometries of

Authors:
Earl Berkson, Robert Kaufman and Horacio Porta

Journal:
Trans. Amer. Math. Soc. **199** (1974), 223-239

MSC:
Primary 47D10; Secondary 46E15

DOI:
https://doi.org/10.1090/S0002-9947-1974-0361923-4

MathSciNet review:
0361923

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Abstract: Let be a strongly continuous one-parameter group of isometries in with unbounded generator. There is a uniquely determined one-parameter group , of Möbius transformations of the (open) disc corresponding to by way of Forelli's theorem. The interplay between and is studied, and the spectral properties of the generator of are analyzed in this context. The nature of the set of common fixed points of the functions plays a crucial role in determining the behavior of . The spectrum of , which is a subset of , can be a discrete set, a translate of or of , or all of . If is not a doubleton subset of the unit circle, can be extended to a holomorphic semigroup of -operators, the semigroup being defined on a half-plane. The treatment of is facilitated by developing appropriate properties of one-parameter groups of Möbius transformations of . In particular, such groups are in one-to-one correspondence (via an initial-value problem) with the nonzero polynomials , of degree at most 2, such that Re for all unimodular . A has an explicit description (in terms of the polynomial corresponding to ) as a differential operator.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1974-0361923-4

Keywords:
Isometry,
,
Möbius transformation,
group,
generator,
spectrum

Article copyright:
© Copyright 1974
American Mathematical Society