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Composition Operators on the Dirichlet Space and Related Problems

Gerardo A. Chacon·Gerardo R. Chacon·Jose Gimenez
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Abstract

In this paper we investigate the following problem: when a bounded analytic function $ϕ$ on the unit disk $\mathbb{D}$, fixing 0, is such that $\{ϕ^n : n = 0, 1, 2, . . . \}$ is orthogonal in $\mathbb{D}$?, and consider the problem of characterizing the univalent, full self-maps of $\mathbb{D}$ in terms of the norm of the composition operator induced. The first problem is analogous to a celebrated question asked by W. Rudin on the Hardy space setting that was answered recently ([3] and [15]). The second problem is analogous to a problem investigated by J. Shapiro in [14] about characterization of inner functions in the setting of $H^2$.

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Gerardo A. Chacon, Gerardo R. Chacon, Jose Gimenez. 2005-04-08. Composition Operators on the Dirichlet Space and Related Problems. https://arxiv.org/abs/math/0504179

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