Marina Levenshtein

The Schwarz Lemma: Rigidity and Dynamics

The Schwarz Lemma has given impetus to developments in several areas of complex analysis and mathematics in general. We survey some investigations related to its three parts (invariance, rigidity, and distortion) that began early in the twentieth century and are still being carried out. We consider only functions analytic in the unit disk. Special attention is devoted to the Boundary Schwarz Lemma and to applications of the Schwarz–Pick Lemma and the Boundary Schwarz Lemma to modern rigidity theory and complex dynamics.

A rigidity theorem for holomorphic generators on the Hilbert ball

AbstractWe present a rigidity property of holomorphic generators on theopen unit ball B of a Hilbert space H. Namely, if f ∈ Hol(B,H)is the generator of a one-parameter continuous semigroup {F t } t≥0 onB such that for some boundary point τ ∈ ∂B, the admissible limitK-lim z→τf(x)kx−τk 3 = 0, then f vanishes identically on B. Let H be a complex Hilbert space with inner product h·,·i and inducednorm k·k. If H is finite dimensional, we will identify H with C n . We denoteby Hol(D,E) the set of all holomorphic mappings on a domain D ⊂ H whichmap D into a subset E of H, and put Hol(D) := Hol(D,D).We are concerned with the problem of finding conditions for a mappingF ∈ Hol(D,E) to coincide identically with a given holomorphic mappingon D, when they behave similarly in a neighborhood of a boundary pointτ ∈ ∂D.Forholomorphicself-mappings oftheopenunitdisk ∆ := {z ∈ C : |z| < 1},thefollowingresult inthisdirection isduetoD.M.Burns andS.G.Krantz[4].Proposition 1. Let F ∈ Hol(∆) . If the unrestricted limit

Growth Estimates for the Numerical Range of Holomorphic Mappings and Applications

The numerical range of holomorphic mappings arises in many aspects of non-linear analysis, finite and infinite dimensional holomorphy, and complex dynamical systems. In particular, this notion plays a crucial role in establishing exponential and product formulas for semigroups of holomorphic mappings, the study of flow invariance and range conditions, geometric function theory in finite and infinite dimensional Banach spaces, and in the study of complete and semi-complete vector fields and their applications to starlike and spirallike mappings, and to Bloch (univalence) radii for locally biholomorphic mappings. In the present paper, we establish lower and upper bounds for the numerical range of holomorphic mappings in Banach spaces. In addition, we study and discuss some geometric and quantitative analytic aspects of fixed point theory, non-linear resolvents of holomorphic mappings, Bloch radii, as well as radii of starlikeness and spirallikeness.

A Burns–Krantz-type Theorem for Pseudo-contractive Mappings

In this note we prove a rigidity result for pseudo-contractive mappings on the unit disk of the complex plane in the spirit of the Burns–Krantz theorem. As an auxiliary assertion, for pseudo-contractive mappings fixing the origin and 1, we establish an inequality connecting the first derivatives at these points.