Podal subspaces on the unit polydisk (Q2773413)

The paper is motivated by the problem of extension of the classical Beurling theorem to higher dimensions. A closed subspace \(M \subset H^2(D^n)\), \(n\geq 1\) is said to be invariant if \(z_iM\subset M\) for \(1\leq i\leq n\). Two such subspaces \(M_1\) and \(M_2\) are called unitarily equivalent (respectively, similar), if there is a unitary (respectively invertible) operator \(T:M_1 \to M_2\) such that \(Tph=pTh\) for all complex polynomials \(p\) and all elements \(h\) in \(M_1\). The unitary (respectively, similarity) orbit for \(M\) is the family of all invariant subspaces unitarily equivalent (respectively, similar) to it. Such an \(M\) is called podal (respectively, \(s\)-podal) if it is largest in its unitary (respectively similarity) orbit.NEWLINENEWLINENEWLINEIn the paper under review, the author studies these concepts and shows, in particular, that some orbits do not contain podal points. Sample result (Proposition 2.18): Let \(M\) be an invariant subspace. Then \(M\) is \(s\)-podal if and only if it is not contained in the kernel of any Hankel operator.