Quotient Hardy modules (Q2705807)

This paper makes interesting contributions to our knowledge of the inner-outer factorization of \(H^2\) functions on the polydisk as well as the rigidity of a certain class of quotient Hardy modules. NEWLINENEWLINENEWLINELet \(D^n\) denote the unit polydisk in \(\mathbb{C}^n\), \(A(D^n)\) the closure in the supremum norm of the polynomials over \(D^n\), and \(H^2(D^n)\) the Hardy space of square-integrable holomorphic functions on the polydisk. For any \(h\) in \(H^2(D^n)\), the \(A(D^n)\)-submodule generated by \(h\) is \([h]:=\overline{A(D^n)h}^{H^2}\) and the associated quotient Hardy module is \(H^2(D^n)\ominus [h]\). In case \([h]=H^2(D^n)\), we say that \(h\) is \textit{outer}. For each \(f\) in \(A(D^n)\), define a bounded linear operator \(S_f\) on \(H^2(D^n)\ominus [h]\) by \(S_f(g)= qfg\), where \(q: H^2(D^n)\to H^2(D^n)\ominus [h]\) is the projection onto the quotient module. In particular, \(S_{z_1},\ldots,S_{z_n}\) are the compressions of the Toeplitz operators \(T_{z_1},\ldots,T_{z_n}\) onto \(H^2(D^n)\ominus [h]\). For brevity, denote \(S_{z_j}\) by \(S_j\). A key observation (Lemma 1.1) is that, if \(\lambda\) is not the \(j\)-th coordinate of any of the zeros of \(h\) in \(\overline{D^n}\), then \(\lambda\) is in the resolvent set of \(S_j\). Together with several results on the joint spectrum of the \(n\)-tuple \((S_{1},\ldots,S_{n})\), this leads to a characterization of the spectrum \(\sigma(S_j)\) in terms of the zero set \(Z(h)\) of \(h\). NEWLINENEWLINENEWLINESpecifically, Theorem 1.5 establishes that, if \(h\) is holomorphic on a neighborhood of \(\overline{D^n}\) and satisfies the condition \(\overline{Z(h)}\cap\overline{D^n}=\overline{Z(h)\cap D^n}\), then \(\sigma(S_j)\) is the projection onto the \(j\)-th coordinate of the set \(\overline{Z(h)\cap D^n}\). Another consequence of Lemma 1.1 is that, if a number \(\lambda\) with \(|\lambda|=1\) is not the \(j\)-th coordinate of any of the zeros of \(h\) in \(\overline{D^n}\), then every function in \(H^2(D^n)\ominus [h]\) has an analytic continuation to a neighborhood of \(D^{j-1}\times \{\lambda\}\times D^{n-j}\). In tandem with a result of Ahern and Clark on analytic continuation, this leads to a nice result on inner-outer factorization. By a \textit{slice} of \(D^n\) we mean a subset of \(D^n\) on which the \(j\)-th coordinate is fixed for some \(j\). NEWLINENEWLINENEWLINETheorem 2.2 states that, if \(h\) is holomorphic in a neighborhood of \(\overline{D^n}\) and satisfies the conditions that (i) \(\overline{Z(h)}\cap\overline{D^n}=\overline{Z(h)\cap D^n}\), (ii) \({Z(h)\cap D^n}\) is not a subset of a countable union of slices of \(D^n\), and (iii) there is an integer \(j\leq n\) such that the projection of \(\overline{Z(h)\cap D^n}\) onto the \(j\)-th coordinate does not contain the unit circle, then \(h\) has no inner-outer factorization.NEWLINENEWLINENEWLINEThe paper concludes with a rigidity theorem (Theorem 3.2) stating that, if \(M_1\) and \(M_2\) are submodules of \(H^2(D^n)\) both generated by bounded holomorphic functions, then \(H^2(D^n)\ominus M_1\) and \(H^2(D^n)\ominus M_2\) are quasi-similar as \(A(D^n)\) modules if, and only if, \(M_1=M_2\). The authors conjecture that this result can be strengthened to apply to the similarity of quotient modules for any submodules of \(H^2(D^n)\).