On an invariant subspace whose common zero set is the zeros of some function. (Q1841602)

Let \({\mathbb D}^n\) be the open unit polydisc in \({\mathbb C}^n\) and \({\mathbb T}^n\) be its distinguished boundary. A closed subspace \(M\) of \(H^2({\mathbb D}^n)\) is said to be invariant if \(z_jM\subset M\) for \(j=1,\ldots,n\). For an invariant subspace \(M\) define \({\mathcal M}(M)=\{\phi\in L^\infty({\mathbb T}^n)\colon\, \phi M\subset H^2({\mathbb D}^n)\}\). \({\mathcal M}(M)\) is called the set of multipliers of \(M\). The common zero set of \(M\) is defined to be \(Z(M)=\{z\in{\mathbb D}^n\colon\, f(z)=0 \;\text{for}\;f\in M\}\). Let \(F\) be a nonzero function in \(H^2({\mathbb D}^n)\) such that for any \(\phi\in L^\infty({\mathbb T}^n)\) from \(\phi F\in H^2({\mathbb D}^n)\) it follows that \(\phi\in H^\infty({\mathbb D}^n)\). The author studies the set \({\mathcal M}(M)\) of an invariant subspace \(M\subset H^2({\mathbb D}^n)\) for which \(Z(M)=Z(F)=\{z\in {\mathbb D}^n \colon\, F(z)=0\}\). The results obtained are generalizations of some results from the author's papers [Can. Math. Bull. 39, 219--226 (1996; Zbl 0864.47001); Arch. Math. 66, 490--498 (1996; Zbl 0856.32002)].