Composition operators on weighted Dirichlet spaces

Abstract

Abstract. We characterize bounded and compact composition operators on weighted Dirichlet spaces. The method involves integral averages of the determining function for the operator, and the connection between composition operators on Dirichlet spaces and Toeplitz operators on Bergman spaces. We also present several examples and counter-examples that point out the borderlines of the result and its connections to other themes.

Compact composition operators on weighted Hilbert spaces of analytic functions

“…Let us recall that Zorboska showed in [10] (see also [3]) that, for ϕ ∈ D α where 0 ≤ α < 1, C ϕ is bounded on D α if and only if N ϕ,α dA(z) is a Carleson measure for A α (D) and C ϕ is compact on D α if and only if N ϕ,α dA(z) is a vanishing Carleson measure for A α (D). More explicitly, for 0 ≤ α < 1 and for all ζ ∈ T, we have…”

The trifecta of Hilbert spaces on Unit Disc

“…From the definition and background section, Section 2 of [26], we see that We see that, undoubtedly all these three Hilbert spaces, A 2 , D 2 and H 2 are connected with each other but in quite complex equalities. One can look up to equalities (4.5) and (4.8) to witness the situation of plugging certain value (here, it is 1) to the parameters (here, they are α and β) to retrieve H 2 from (weighted) Bergman and (weighted) Dirichlet space in respective sense.…”

Comparison of singular numbers of composition operators on different Hilbert spaces of analytic functions

“…The image Ω of the (univalent) cusp map χ is formed by the intersection of the inside of the disk D 1 − a 2 , a 2 and the outside of the two closed disks D 1 + ia 2 , a 2 and D 1 − ia 2 , a 2 . Since χ is injective, it follows from Zorboska's characterization in [44] (see also [36,Section 6.12]) that the composition operator C χ is bounded on D 2 α for α ≥ 0. In particular χ ∈ D 2 α .…”