Invariant subspaces and operator model theory on noncommutative varieties

Abstract

Let ${{\mathcal {Q}}}\subset {{\mathbb {C}}}\left<Z_1,\ldots , Z_n\right>$ be an arbitrary set of polynomials in noncommutative indeterminates such that $q(0)=0$ for all $q\in {{\mathcal {Q}}}$. The noncommutative variety

$$\begin{aligned} {{\mathcal {V}}}_{f,{{\mathcal {Q}}}}^m({{\mathcal {H}}}):=\left\{ { X}=(X_1,\ldots , X_n)\in \mathbf{D}_f^m({{\mathcal {H}}}): \ q({ X})=0 \ \text { for all } q\in {{\mathcal {Q}}}\right\} , \end{aligned}$$

where $\mathbf{D}_f^m({{\mathcal {H}}})$ is a noncommutative regular domain in $B({{\mathcal {H}}})^n$ and $B({{\mathcal {H}}})$ is the algebra of bounded linear operators on a Hilbert space ${{\mathcal {H}}}$, admits a universal model $B^{(m)}=(B_1^{(m)},\ldots , B_n^{(m)})$ such that $q(B^{(m)})=0$, $q\in {{\mathcal {Q}}}$, acting on a model space which is a subspace of the full Fock space with n generators. In this paper, we obtain a Beurling type characterization of the joint invariant subspaces under the operators $B_1^{(m)},\ldots , B_n^{(m)}$, in terms of partially isometric multi-analytic operators acting on model spaces. More generaly, a Beurling-Lax-Halmos type representation is obtained and used to parameterize the wandering subspaces of the joint invariant subspaces under $B_1^{(m)}\otimes I_{{\mathcal {E}}},\ldots , B_n^{(m)}\otimes I_{{\mathcal {E}}}$, and to characterize when they are generating for the corresponding invariant subspaces. Similar results are obtained for any pure n-tuple $(X_1,\ldots , X_n)$ in the noncommutative variety ${{\mathcal {V}}}_{f,{{\mathcal {Q}}}}^m({{\mathcal {H}}})$. We characterize the elements in the noncommutative variety ${{\mathcal {V}}}_{f,{{\mathcal {Q}}}}^m({{\mathcal {H}}})$ which admit characteristic functions, develop an operator model theory for the completely non-coisometric elements, and prove that the characteristic function is a complete unitary invariant for this class of elements. This extends the classical Sz.-Nagy–Foiaş functional model for completely non-unitary contractions, based on characteristic functions. Our results apply, in particular, when ${{\mathcal {Q}}}$ consists of the noncommutative polynomials $Z_iZ_j-Z_jZ_i$, $i,j=1,\ldots , n$. In this case, the model space is a symmetric weighted Fock space, which is identified with a reproducing kernel Hilbert space of holomorphic functions on a Reinhardt domain in ${{\mathbb {C}}}^n$, and the universal model is the n-tuple $(M_{\lambda _1},\ldots , M_{\lambda _n})$ of multipliers by the coordinate functions.