On the Ideal Structure of the Nevanlinna Class

Abstract

Abstract.Let N denote the Nevanlinna class, i.e. the algebra of holomorphic functions of bounded characteristic in the open unit disc. We study analytic conditions for a finitely generated ideal to be equal to the whole algebra N. Then we characterize the finitely generated prime ideals containing a nontangential interpolating Blaschke product. Further, we give an example of an ideal of N whose closure in the natural metric on A' is not an ideal.

Finitely generated ideals in the Nevanlinna class

“…Since |B * * 1 (α j )| ≤ |B * * 2 (α j )| and ρ(α * j , α j ) = |B * * 1 (α j )|/4, Schwarz' Lemma (see Lemma 2.1(a)) gives |B * * 2 (α * j )| ≥ |B * * 2 (α j )|/2 and |B * * 1 (α * j )| ≤ 3|B * * 1 (α j )|/2. Hence, using again Lemma 2.1(a) and (4.14), 16) where C 2 > 0 is an absolute constant. From (4.15) and (4.16) we get…”

“…For the Nevanlinna class R. Mortini observed that a well known result of T. Wolff implies the following corona theorem (see [17] or [16]).…”

The Corona Property in Nevanlinna quotient algebras and interpolating sequences

“…has a harmonic majorant (the function H). Therefore, by (13), the map λ k −→ M k k ∈ ∪ j J j has no harmonic majorant. This is (b) in the statement.…”

“…The Corona problem for the Nevanlinna class was solved by R. Mortini in [12, Satz 4] (see also [13]). Let Har + (D) denote the cone of positive functions in the unit disk.…”

From $ H^\infty$ to N. Pointwise properties and algebraic structure in the Nevanlinna class

“…For the Nevanlinna class R. Mortini observed that a well-known result of T. Wolff implies the following corona theorem (see [32] or [26]). Theorem 5.2.…”