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

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 $\mathbf{\text{Hardy}}\mathbf{\text{type}}$ 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.