A note on connectedness of Blaschke products

TL;DR

This paper investigates the topological properties of a specific class of Blaschke products within the space of inner functions, revealing their connectedness and factorization characteristics under the uniform topology.

Contribution

Introduces the class H_{SC} of Blaschke products and proves their path-connectedness with their multiples, providing new insights into their zero sequences and factorization properties.

Findings

Blaschke products in H_{SC} are path-connected to their multiples.

Each H_{SC} Blaschke product has an interpolating, one-component factor.

The results aid in selecting fine subsequences of zeros for these products.

Abstract

Consider the space $\mathcal{F}$ of all inner functions on the unit open disk under the uniform topology, which is a metric topology induced by the $H^{\infty}$-norm. In the present paper, a class of Blaschke products, denoted by $\mathcal{H}_{SC}$, is introduced. We prove that for each $B\in\mathcal{H}_{SC}$, $B$ and $zB$ belong to the same path-connected component of $\mathcal{F}$. It plays an important role of a method to select a fine subsequence of zeros. As a byproduct, we obtain that each Blaschke product in $\mathcal{H}_{SC}$ has an interpolating and one-component factor.