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spaces coincide with the Bloch space for
and are subspaces of
. We obtain lower and upper estimates for the essential norm of a composition operator from the Bloch space into
, in particular from the Bloch space into
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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