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We study when multiplication by a weight can turn a non-compact compositionoperator on H 2 into a compact operator, and when it can be in Schattenclasses. The q-summing case in H p is considered. We also study when thismultiplication can turn a compact composition operator into a non-compact one.MSC 2010 primary: 47B33 ; secondary: 46B28

Compactification, and beyond, of composition operators on Hardy spaces by weights