Optimal target de Branges–Rovnyak space for composition operators

Identify an optimal de Branges–Rovnyak space containing the range of the composition operator C_φ: 𝓗(b) → 𝓗(η) for general b, φ ∈ H^∞₁, where η is one of the symbols constructed in the paper, and characterize the corresponding minimality property.

Background

The paper establishes several general inclusions of the form C_φ(𝓗(b)) ⊆ 𝓗(η), with η determined by b and φ. In special cases involving inner functions, the target space can satisfy a minimality property, but the examples in the final section show that the target spaces produced by the general theorems may be far from optimal.

For specific choices of b and φ, the authors exhibit target spaces that are incomparable with 𝓗(b), while their intersection can yield a strictly smaller de Branges–Rovnyak space. These examples motivate determining whether an optimal target space always exists and, if so, identifying it. The problem is technically difficult because although inclusion between general de Branges–Rovnyak spaces has been characterized, the available criterion is highly technical.