One-component inner functions

Abstract

Definition 1 An inner function u in H ∞ is said to be a one-component inner function if there is η ∈]0, 1[ such that the level set (also called sublevel set or filled level set) � u (η) := {z ∈ D : |u(z)| < η} is connected.One-component inner functions, the collection of which we denote by I c , were first studied by Cohn [10] in connection with embedding theorems and Carleson measures. It was shown that [for instance, [10], p. 355] arclength on {z ∈ D : |u(z)| = ε} is such a measure whenever is connected and η < ε < 1.A thorough study of the class I c was given by Aleksandrov [1] who showed the interesting result that u ∈ I c if and only if there is a constant C = C(u) such that for all a ∈ D Many operator-theoretic applications are given in [1][2][3]7]. In our paper here, we are interested in explicit examples, which are somewhat lacking in the literature. For example, if S is the atomic inner function, which is given by AbstractWe explicitly unveil several classes of inner functions u in H ∞ with the property that there is η ∈]0, 1[ such that the level set � u (η) := {z ∈ D : |u(z)| < η} is connected. These so-called one-component inner functions play an important role in operator theory.

Remarks on one-component inner functions

“…As a consequence of Theorem 1, we obtain the affirmative answer to the following question posed in [13]: Is the Blaschke product B with zeros z n = 1 − n −2 for n ∈ N a one-component inner function? Some other examples of one-component inner functions are listed below.…”

“…As a concrete example, we mention that the Blaschke product with zeros z n = 1−2 −n for n ∈ N is a one-component inner function [13]. In addition, it is a well-known fact that every finite Blaschke product is a one-component inner function.…”

“…We interpret that finite Blaschke products are not thin. By [13,Corollary 21], any thin Blaschke product is not a one-component inner function. Using this fact and [12, Proposition 4.3(i)], we can give an example which shows that condition (1.1) in Theorem 1 is essential.…”

“…An atomic singular inner function associated with a measure having only finitely many mass points is a one-component inner function; see [13,Corollary 17]. In the literature, one cannot find any example of a one-component singular inner function associated with a measure having infinitely many mass points.…”

A characterization of one‐component inner functions

“…As a byproduct he proved also a strong form of the Schwarz–Pick lemma for inner functions in . Using Aleksandrov's descriptions, Cima, Mortini and the second author constructed some concrete examples of one‐component inner functions [10, 11, 23]. In particular singular inner functions associated to a finite sum of weighted Dirac masses are one‐component and thin Blaschke products are not in .…”