Ph.D Thesis
Ph.D Student Levenshtein Marina Rigidity Theory for Holomorphic Mappings Department of Mathematics Professor Emeritus Simeon Reich Professor David Shoikhet

Abstract

In this thesis we study rigidity properties of some classes of holomorphic mappings on the open unit disk of the complex plane and on the open unit ball of a complex Hilbert space .

Our results may be divided into three parts.

1.  We study the problem of finding conditions for holomorphic mappings on to coincide identically with a given holomorphic mapping , when they behave similarly in a neighborhood of a boundary point .
1. We consider the class of functions which satisfy the inequality , ,for some , and establish sufficient conditions at a boundary point for functions of this class to vanish identically on .
2. We consider rigidity properties of the holomorphic generators of one-parameter continuous semigroups on . We show that for a holomorphic generator on , the equality for some boundary point implies that on . Moreover, we prove that it is even sufficient to assume that there exists a sequence? converging nontangentially to such that , to conclude that on .
3. We show that for the holomorphic generator of a one-parameter continuous semigroup on the open unit ball of a complex Hilbert space , the equality for some boundary point implies that on . We also present an improvement of this result for the weak restricted -limit of .
2. We investigate identity principles for commuting holomorphic self-mappings of theunit disk . Our result in this direction? asserts that if one of two commuting holomorphic self-mappings of is of hyperbolic type, then so is the second one (if it is not the identity).
3. We study rigidity properties of two semigroups ?and such that for all , that is, commuting semigroups. We first establish a criterion for two semigroups to commute in terms of their generators. As a consequence of this result, we obtain a rigidity theorem which gives conditions on the generators and of two commuting semigroups and which imply that the semigroups coincide.

We also consider the following problem: does the commutativity of only two elements and of the semigroups and imply that the semigroups commute? We prove that in the case of hyperbolic semigroups this question has an affirmative answer. For semigroups with an interior fixed point and for semigroups of parabolic type the answer is negative in general. In these cases we formulate some supplementary conditions which

ensure the commutativity of the semigroups.