Invariant subspaces and operator model theory on noncommutative varieties

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