Invariant subspaces for Bishop-type operators

Abstract

Nontrivial hyperinvariant subspaces are exhibited for a class of weighted translation operators which includes many operators of the form M φ U α , on L 2[0, 1), where φ ϵ L α [0, 1) and U α , is translation, modulo one, by an irrational α. In particular, if φ is the restriction to [0, 1) of a function analytic in some open neighbourhood of [0, 1] in the complex plane, then M φ U α has a nontrivial hyperinvariant subspace for almost all α.