Automatic differentiability and characterization of cocycles of holomorphic flows

Jafari, Farhad; Tonev, Thomas; Toneva, Elena

doi:10.1090/S0002-9939-05-07904-9

Automatic differentiability and characterization of cocycles of holomorphic flows
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by Farhad Jafari, Thomas Tonev and Elena Toneva PDF

Proc. Amer. Math. Soc. 133 (2005), 3389-3394 Request permission

Abstract:

In this paper we prove that cocycles of holomorphic flows on domains in the complex plane are automatically differentiable with respect to the flow parameter, and their derivatives are holomorphic functions. We use this result to show that, on simply connected domains, an additive cocycle is a coboundary if and only if this cocycle vanishes at the fixed point of the flow.

References

Earl Berkson and Horacio Porta, Semigroups of analytic functions and composition operators, Michigan Math. J. 25 (1978), no. 1, 101–115. MR 480965
Carl C. Cowen, Subnormality of the Cesàro operator and a semigroup of composition operators, Indiana Univ. Math. J. 33 (1984), no. 2, 305–318. MR 733903, DOI 10.1512/iumj.1984.33.33017
J.-Cl. Evard and F. Jafari, On semigroups of operators on Hardy spaces, preprint (1995).
F. Jafari, T. Tonev, E. Toneva, and K. Yale, Holomorphic flows, cocycles, and coboundaries, Michigan Math. J. 44 (1997), no. 2, 239–253. MR 1460412, DOI 10.1307/mmj/1029005702
Krzysztof Jarosz, Automatic continuity of separating linear isomorphisms, Canad. Math. Bull. 33 (1990), no. 2, 139–144. MR 1060366, DOI 10.4153/CMB-1990-024-2
Wolfgang König, Semicocycles and weighted composition semigroups on $H^p$, Michigan Math. J. 37 (1990), no. 3, 469–476. MR 1077330, DOI 10.1307/mmj/1029004204
Aristomenis G. Siskakis, Composition semigroups and the Cesàro operator on $H^p$, J. London Math. Soc. (2) 36 (1987), no. 1, 153–164. MR 897683, DOI 10.1112/jlms/s2-36.1.153