Compact Composition Operators on the Bloch Space

Abstract

Abstract.Necessary and sufficient conditions are given for a composition operator C^f = f o <f> to be compact on the Bloch space & and on the little Bloch space 38$ . Weakly compact composition operators on £ § § are shown to be compact. If <t> 6 3 § § is a conformai mapping of the unit disk D into itself whose image <¡>(H) approaches the unit circle T only in a finite number of nontangential cusps, then C$ is compact on 3g § . On the other hand if there is a point of T n <p(H>) at which <£(D) does not have a cusp, then C¿ is not compact.

Some new properties of composition operators associated with lens maps

“…Let us quote some examples: -X = H 1 ; this was proved by D. Sarason in 1990 ([20]); -X = H ∞ and the disk algebra X = A(D) (A. Ülger [23] and R. Aron, P. Galindo and M. Lindström [1], independently; the first-named of us also gave another proof in [7]); -X is the little Bloch space B 0 (K. Madigan and A. Matheson [17]); -X is the Hardy-Orlicz spaces X = H ψ , when the Orlicz function ψ grows more rapidly than power functions, namely when it satisfies the condition ∆ 0 ( [11], Theorem 4.21, page 55); -X = BM OA and X = V M OA (J. Laitila, P. J. Nieminen, E. Saksman and H.-O. Tylli [6]).…”

Sums of Weighted Composition Operators on Cop

“…We denote by S(‫)ބ‬ the set of analytic self-maps of ‫.ބ‬ For u ∈ H ∞ and ϕ ∈ S(‫,)ބ‬ we may define the weighted composition operator uC ϕ on H ∞ by (uC ϕ )f = u(f • ϕ) for f ∈ H ∞ . It is known that if ϕ ∈ QA, then C ϕ maps QA into QA (see [15]), and if ϕ ∈ COP, then C ϕ maps COP into COP (see [9]). …”

Weighted composition operators on the predual and dual Banach spaces of Beurling algebras onZ

“…The first paper dealing with boundedness and compactness of composition operators on spaces isomorphic to c 0 or l ∞ traces back to the work of Madigan and Matheson [7]. This work was later refined by Montes-Rodríguez [8,9].…”