Cascade Connections of Linear Systems and Factorizations of Holomorphic Operator Functions Around a Multiple Zero in Several Variables

We show that the factorization problem $\theta (z)=\theta_2(z)\theta_1(z)$ is solvable in the class of Hilbert space operator-valued functions holomorphic on some neighbourhood of $z=0$ in $\nspace{C}{N}$ and having a zero at $z=0$ (here $\theta (z)$ has a multiple zero at $z=0$). Such a factorization problem becomes more complicated if we demand for $\theta (z), \theta_1(z)$ and $\theta_2(z)$ to be Agler--Schur-class functions on the polydisk $\nspace{D}{N}$ and for the factorization identity to hold in $\nspace{D}{N}$. In this case we reduce it to the problem on the existence of a cascade decomposition for certain multiparametric linear system $\alpha$--a conservative realization of $\theta (z)$, and give the criterion for its solvability in terms of common invariant subspaces for the $N$-tuple of main operators of $\alpha$.