Complex structure that admits complete Nevanlinna-Pick spaces of Hardy type

Abstract

In this paper, we will characterize those sets, over which every irreducible complete Nevanlinna--Pick space enjoys that its multiplier and supremum norms coincide. Moreover, we will prove that, if there exists an irreducible complete Nevanlinna--Pick space of holomorphic functions on a reduced complex space $X$ whose multiplier algebra is isometrically equal to the algebra of bounded holomorphic functions (we will say that such a space is of $\textbf{Hardy type}$ in this paper), then $X$ must be biholomorphic to the unit disk minus a zero analytic capacity set. This means that the Hardy space is characterized as a unique irreducible complete Nevanlinna--Pick space of Hardy type.