Invariant subspaces for Bishop-type operators
The main focus of this paper is hyperinvariant subspaces for weighted translation operators on \(L^ 2[0,1]\). The author exhibits a rather large class including many operators of the form \(M_{\phi}U_{\alpha}\) on \(L_ 2[0,1]\), where \(\phi \in L^{\infty}[0,1]\) and \(U_{\alpha}\) is translation modulo one, by an irrational \(\alpha\). If for example \(\phi\) is the restriction to [0,1] of a function analytic in some open neighborhood of [0,1] in the complex plane, then \(M_{\phi}U_{\alpha}\) has a nontrivial hyperinvariant subspace for almost all \(\alpha\). This improves a result of \textit{A. M. Davie} [Bull. London Math. Soc. 6, 343- 348 (1974; Zbl 0287.47003)]. Originally operators of type \(M_ xU_{\alpha}\) have been suggested by Bishop as candidates for operators with no nontrivial subspace. The question however of whether \(M_ xU_{\alpha}\) has a nontrivial invariant subspace for all \(\alpha\) remains open.
hyperinvariant subspaces for weighted translation operators on \(L^ 2[0,1]\)
scientific article; zbMATH DE number 4200967
scientific article