Title | RIGIDITY OF HOLOMORPHIC GENERATORS AND ONE-PARAMETER SEMIGROUPS |
Publication Type | Journal Article |
Year of Publication | 2007 |
Authors | ELIN MARK, LEVENSHTEIN MARINA, SHOIKHET DAVID, TAURASO ROBERTO |
Journal | Dynamic Systems and Applications |
Volume | 16 |
Start Page | 251 |
Pagination | 16 |
Date Published | 2007 |
ISSN | 1056-2176 |
AMS Subject Classification | 30D05, 32H12, 47B33, 47H20 |
Abstract | In this paper we establish a rigidity property of holomorphic generators by using their local behavior at a boundary point τ of the open unit disk ∆. Namely, if f ∈ Hol(∆, C) is the generator of a one- parameter continuous semigroup {Ft }t ≥ 0, we show that the equality f(z) = o (|z − τ |3) when z → τ in each non-tangential approach region at τ implies that f vanishes identically on ∆. Note, that if F is a self-mapping of ∆ then f = I − F is a generator, so our result extends the boundary version of the Schwarz Lemma obtained by D. Burns and S. Krantz. We also prove that two semigroups { Ft }t ≥ 0 and { Gt }t ≥ 0, with generators f and g respectively, commute if and only if the equality f = αg holds for some complex constant α. This fact gives simple conditions on the generators of two commuting semigroups at their common null point τ under which the semigroups coincide identically on ∆. |
https://acadsol.eu/dsa/articles/16/251-266-DSA-26-08-Elin.pdf | |
Refereed Designation | Refereed |