Compact composition operators on $H^p(B_N)$.

Carleson Measures in Hardy and Weighted Bergman Spaces of Polydiscs

“…Hastings [4] has given a similar result for unweighted Bergman spaces. Recently MacCluer [7] has obtained a Carleson measure characterization of identity operators on Hardy spaces of the unit ball of C" using the well-known results of Hormander, and Stegenga [10] has considered similar questions in the unit disc. In this work, which is inspired by the results of Chang and Hastings, we present three theorems which characterize the compact identity operators from Hardy spaces of polydiscs into Lp spacfes, and the corresponding bounded and compact identity operators on weighted Bergman spaces.…”

Composition operators on Dirichlet-type spaces

“…Thus the condition Φ ∞ < 1 is not necessary for C Φ to be compact on the Dirichlet-type spaces D p . In comparison, MacCluer proved that C Φ is compact on S p (1 ≤ p < ∞) if and only if Φ ∈ S p and Φ ∞ < 1 [14]. Proof.…”

“…Interest in the spaces D p is motivated by the work of R. Roan [19] and B. D. MacCluer [14], who studied composition operators on S p , the space of functions with derivatives in the Hardy space H p for p ≥ 1.…”

“…Carleson measures have been useful in the study of composition operators in several settings [11,13,14,15,27,28]. For w ∈ D, let N 2 (Φ, w) denote the number of zeroes (counting multiplicities) of the equation Φ(z) − w. For 1 ≤ p < 2 and w ∈ D, N p (Φ, w) is defined to be the modified counting function…”

Composition Operators on Finite Rank Model Subspaces

“…The compactness of C ϕ on Hardy spaces is also studied in [13]. There is also another characterisation for compactness in [9]. Composition operators on many other spaces have been extensively studied by many authors.…”