Shift invariant subspaces with arbitrary indices in \(\ell^p\) spaces
Let \(S(a_0,a_1,\ldots)=(0,a_0,a_1,\ldots)\) be the shift operator on \(\ell_2\). It follows from Beurling's theorem that \(\dim (M/SM)=1\) for every non-zero shift invariant subspace \(M\) of \(\ell_2\). In this paper, the authors consider \(S\) as an operator on some weighted \(\ell_p\) spaces, \(0<p<\infty\). The index of the invariant subspace \(M\) is \(\dim (M/\overline{SM})\). It is shown that, for \(2<p<\infty\) and for every \(n=2,3,\ldots,\infty\), there exist shift invariant subspaces of \(\ell_p\), or more generally of some weighted \(\ell_p\) spaces, with index \(n\). The proof is rather long and technical, and the authors begin with the case \(n=2\), without weight, which is simpler. They point out that the holomorphic functions naturally associated to the elements of the shift invariant subspaces they construct have no common zeros in the unit disk \({\mathcal D}\). The paper ends with several open questions.
scientific article; zbMATH DE number 1763765