Hankel operators, invariant subspaces, and cyclic vectors in the Drury-Arveson space (Q2796727)

Let \(\mathbb B_d\) (\(d\geq 1\)) be the unit ball in the space \(\mathbb C^d\). The Drury-Arveson space \(H_d^2\) is the Hilbert space of holomorphic functions \(f:\mathbb B_d\to\mathbb C\) which have the power expansion \(f(x)=\sum_\alpha c_\alpha z^\alpha\) such that \(\| f \|^2=\sum_\alpha| c_\alpha|^2\frac{\alpha!}{|\alpha|!}<\infty\). Here, for the multi-index \(\alpha=(\alpha_1,\dots,\alpha_d)\) \(\alpha!=\alpha_1!\dots\alpha_d!\), \(|\alpha|=|\alpha_1|+\dots+|\alpha_d|\). It is proved that every nonzero invariant subspace of the \(d\)-tuple \(M_z=(M_{z_1},\dots,M_{z_d})\) is the intersection of kernels of little Hankel operators. This result is used to prove the following assertion: if \(f\), \(1/f\in H_d^2\) then \(f\) is cyclic in \(H_d^2\).