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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Motivated by a formula of P. Jones that gives the distance of a Bloch function to BMOA, the space of bounded mean oscillations, we obtain several formulas for the distance of a Bloch function to the little Bloch space, B o . Immediate consequences are equivalent expressions for functions in Bo-We also give several examples of distances of specific functions to Bo. We comment on connections between distance to Bo and the essential norm of some composition operators on the Bloch space, B. Finally we show that the distance formulas in B have Bloch type spaces analogues.

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Let φ \varphi be an analytic self-map of the open unit disk D \mathbb {D} . We study the complex symmetry of composition operators C φ C_\varphi on weighted Hardy spaces induced by a bounded sequence. For any analytic self-map of D \mathbb {D} that is not an elliptic automorphism, we establish that if C φ C_{\varphi } is complex symmetric, then either φ ( 0 ) = 0 \varphi (0)=0 or φ \varphi is linear. In the case of weighted Bergman spaces A α 2 A^{2}_{\alpha } , we find the non-automorphic linear fractional symbols φ \varphi such that C φ C_{\varphi } is complex symmetric.