A note on connectedness of Blaschke 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.

25 pages, 3 figures