Hardy-space function theory, operator model theory, and dissipative linear systems: the multivariable, free-noncommutative, weighted Bergman-space setting (1906.02814v2)

Abstract: It is known that (i) a subspace ${\mathcal N}$ of the Hardy space $H^2$ which is invariant under the backward shift operator can be represented as the range of the observability operator of a conservative discrete-time linear system, (ii) the transfer-function of this conservative linear system in turn is the inner Beurling-Lax representer for the forward-shift invariant subspace ${\mathcal M} : = {\mathcal N}^\perp$, and (iii) this transfer function also serves as the Sz.-Nagy-Foias characteristic function of the pure contraction operator $T$ given by $T = P_{\mathcal N} M_z |_{\mathcal N}$. The main focus of this paper is to present the extension of this structure to a more general setting. The Hardy space is replaced by the full weighted Bergman-Fock space of formal power series in $d$ freely noncommutative indeterminates, where the shift is replaced by the right shift tuple, where the conservative/dissipative discrete-time linear system becomes a certain type of conservative/dissipative multidimensional linear system with time-varying weights and with evolution along a rooted tree with each node having $d$ forward branches, where a backward shift-invariant subspace ${\mathcal N}$ is the range of the observability operator for such a weighted-Bergman multidimensional linear system, and where the transfer function of this system is the Beurling-Lax representer for the forward shift-invariant subspace ${\mathcal M} = {\mathcal N}^{[\perp]}$, and where this transfer function also serves as the characteristic function for the operator tuple having hypercontractive-operator-tuple adjoint equal to the restriction of the backward-shift tuple to $\mathcal N$.