Quasi-wandering subspaces in a class of reproducing analytic Hilbert spaces (Q2846849)

Let \(H\) be a Hilbert space of analytic functions on the unit ball in \(\mathbb{C}^d\) for some \(d \geq 1\). For \(j=1,\dots, d\), we denote by \(S_j\) the operator of multiplication by the coordinate function \(z_j\). Put \(\bar{S}=(S_1,\dots, S_d)\). Recall that a closed subspace \(M\) of \(H\) is an invariant subspace for \(\bar{S}\) if \(S_jM\subset M\) for all \(j=1,\dots, d\). For such a subspace, set NEWLINE\[NEWLINEP_{M}\bar{S}M^{\perp} = P_MS_1M^{\perp}+\dots+P_MS_dM^{\perp}.NEWLINE\]NEWLINE Here as usual, \(M^{\perp}\) denotes the orthogonal complement of \(M\) in \(H\) and \(P_M\) is the orthogonal projection from \(H\) onto \(M\). We say that \(\bar{S}\) has the \textit{quasi-wandering property} if any invariant subspace \(M\) coincides with the invariant subspace generated by \(P_M\bar{S}M^{\perp}\).NEWLINENEWLINEThe paper under review studies the quasi-wandering property of \(\bar{S}\) on reproducing kernel Hilbert spaces whose kernels \(k\) are unitary invariant. It is shown that, if \(k\) is either a complete Nevanlinna-Pick kernel or a kernel of the form \((1-\langle z,w\rangle)^{-v}\) for some constant \(v>0\), then \(\bar{S}\) has the quasi-wandering property. This result generalizes known results on the Hardy and Bergman spaces over the unit disk.NEWLINENEWLINEThe paper ends with the question whether \(\bar{S}\) always has the quasi-wandering property on \textit{all} analytic reproducing Hilbert spaces defined by unitary invariant kernels.