Erratum to: Closed-Range Composition Operators on $${\mathbb{A}^2}$$ and the Bloch Space

Abstract

We thank Nina Zorboska for pointing out an error in the proof of [2, Proposition 3.1]; a faux pas in our application of Julia’s Theorem. Indeed, we find that Proposition 3.1 does not hold in general (see Example 2.4 in this erratum). This impacts two operator-theoretic results in Section 3 of [2]; namely, Theorems 3.5 and 3.7. Theorem 3.5 (in [2]) does not hold in general. While statements (i) and (ii) of Theorem 3.7 are equivalent (cf. [1, Theorem 2.4]), statement (iii) implies, but is not a consequence of (i) and (ii) (in general). Both of these theorems hold under an additional hypothesis. We make all of this clear in our work here. And, although [2, Corollary 3.6] was established using Theorem 3.5, we find that this corollary holds in general; see the proof of Corollary 1.6 in this erratum. In order that our presentation be somewhat self-contained, we begin by reviewing the terminology of [2]. Let D denote the unit disk {z : |z| < 1} and let T denote its boundary {z : |z| = 1}. Let A denote normalized two-dimensional Lebesgue measure on D and let m denote normalized Lebesgue measure on T; normalized so that these are probability measures. The Bergman space A is the Hilbert space of functions f that are analytic in D such that