arXiv subjects

Joseph Cima

Publications and source records attributed to Joseph Cima.

3 recordsLinked to original sources

arXiv2018

One-component inner functions II

We continue our study of the set $\mathfrak I_c$ of inner functions $u$ in $H^\infty$ with the property that there is $\eta\in ]0,1[$ such that the level set $\Omega_u(\eta):=\{z\in\mathbb D: |u(z)|<\eta\}$ is connected. These functions are called one-component inner functions. They play an important role in function theory and operator theory. Here we show that the composition of two one-component inner functions is again in $\mathfrak I_c$. We also give conditions under which a factor of one-component inner function belongs to $\mathfrak I_c$.

math.CV↗

arXiv2017

One-component inner functions

We explicitely unveil several classes of inner functions $u$ in $H^\infty$ with the property that there is $\eta\in ]0,1[$ such that the level set $\Omega_u(\eta):=\{z\in\mathbb D: |u(z)|<\eta\}$ is connected. These so-called one-component inner functions play an important role in operator theory.

math.CV↗

arXiv2007

The Caratheodory-Cartan-Kaup-Wu theorem on an infinite dimensional Hilbert space

This paper treats a holomorphic self-mapping f: Omega --> Omega of a bounded domain Omega in a separable Hilbert space H with a fixed point p. In case the domain is convex, we prove an infinite-dimensional version of the Cartan-Caratheodory-Kaup-Wu Theorem. This is basically a rigidity result in the vein of the uniqueness part of the classical Schwarz lemma. The main technique, inspired by an old idea of H. Cartan, is iteration of the mapping f and its derivative. A normality result for holomorphic mappings in the compact-weak-open topology, due to Kim and Krantz, is used.

math.CV↗