Abstract. We give a Carleson measure characterization of the compact composition operators on Besov spaces. We use this characterization to show that every compact composition operator on a Besov space is compact on the Bloch space. Finally we give conditions that guarantee that the converse holds.

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For any analytic self-map ϕ of {z : |z| < 1} we give four separate conditions, each of which is necessary and sufficient for the composition operator Cϕ to be closed-range on the Bloch space B. Among these conditions are some that appear in the literature, where we provide new proofs. We further show that if Cϕ is closed-range on the Bergman space A 2 , then it is closed-range on B, but that the converse of this fails with a vengeance. Our analysis involves an extension of the Julia-Carathéodory Theorem. Mathematics Subject Classification (2010). Primary 47B33, 47B38; Secondary 30D55.Let D denote the unit disk {z : |z| < 1} and let T denote the unit circle {z : |z| = 1}. We let A denote two-dimensional Lebesgue measure on D. The Bergman space A 2 is the collection of functions f that are analytic in D such thatAs a closed subspace of L 2 (A), A 2 forms a Hilbert space with respect to the inner product := D f gdA. The Bloch space B is the collection of functions f that are analytic in D such thatNow ||·|| B defines a norm on B, and under this norm B forms a Banach space. Moreover, ||f || A 2 ≤ 3||f || B for any function f that is analytic in D, and hence B ⊆ A 2 . A function ϕ that is analytic in D and that satisfies ϕ(D) ⊆ D is

We characterise the compact composition operators from any Mobius invariant Banach space to VMOA, the space of holomorphic functions on the unit disk U that have vanishing mean oscillation. We use this to obtain a characterisation of the compact composition operators from the Bloch space to VMOA. Finally, we study some properties of hyperbolic VMOA functions. We show that a function is hyperbolic VMOA if and only if it is the symbol of a compact composition operator from the Bloch space to VMOA.

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