Trace ideal criteria for embeddings and composition operators on model spaces

Abstract

Let K ϑ be a model space generated by an inner function ϑ. We study the Schatten class membership of embeddings I : K ϑ ֒→ L 2 (µ), µ a positive measure, and of composition operators Cϕ :In the case of onecomponent inner functions ϑ we show that the problem can be reduced to the study of natural extensions of I and Cϕ to the Hardy-Smirnov space E 2 (D) in some domain D ⊃ D. In particular, we obtain a characterization of Schatten membership of Cϕ in terms of Nevanlinna counting function. By example this characterization does not hold true for general ϑ.

One-component inner functions

“…n , η) can meet at the most at one disk D ρ (z (2) m , η) for n large. Hence where the set on the right-hand side obviously is disconnected.…”

“…It was shown that [for instance, [10], p. 355] arclength on {z ∈ D : |u(z)| = ε} is such a measure whenever is connected and η < ε < 1.A thorough study of the class I c was given by Aleksandrov [1] who showed the interesting result that u ∈ I c if and only if there is a constant C = C(u) such that for all a ∈ D Many operator-theoretic applications are given in [1][2][3]7]. In our paper here, we are interested in explicit examples, which are somewhat lacking in the literature.…”

A characterization of one‐component inner functions

“…See [20]. The class also appears naturally in several recent results in the context of operator theory in [5–8].…”

Duality theorems for coinvariant subspaces of H1

“…Conversely, take a function b ∈ BMO(σ α ) and consider the functional Φ b densely defined on K 1 θ ∩ zH 1 by formula (4). For every σ α -atom a supported on an arc ∆ we have…”