Beurling-Lax representation for weighted Bergman-shift invariant subspaces

The Beurling-Lax-Halmos theorem tells us that any invariant subspace ${\mathcal M}$ for the shift operator $S \colon f(z) \mapsto z f(z)$ on the vectorial Hardy space over the unit disk $H^{2}_{\mathcal Y} = \{f(z) = \sum_{j=0}^{\infty} f_{j} z^{j} \colon \| f \|^{2} =\sum_{j \ge 0} \| f_{j} \|^{2} \u3c \infty\}$ (the Reproducing Kernel Hilbert Space with reproducing kernel $K(z,w) = (1 - z \overline{w})^{-1} I_{\mathcal Y}$) can be represented as ${\mathcal M} = M_{\Theta} H^{2}_{\mathcal U}$ where $M_{\Theta} \colon H^{2}_{\mathcal U} \to H^{2}_{\mathcal Y}$ is an isometric multiplication operator $M_{\Theta} \colon u(z) \mapsto \Theta(z) u(z)$. We focus on three constructions of $\Theta(z)$: \smallskip \noindent (1) \textbf{the wandering subspace construction of Halmos}: ${\mathcal M} = \bigoplus_{j \ge 0} S^{j} ({\mathcal M} \ominus S {\mathcal M}) = M_{\Theta} H^{2}_{\mathcal U}$ where $M_{\Theta}$ is constructed so that $M_{\Theta} \colon \cU \to {\mathcal M} \ominus S {\mathcal M}$ is unitary; \smallskip \noindent (2) as the \textbf{Sz.-Nagy--Foias characteristic function} $\Theta_{T}(z)$ of the $C_{\cdot 0}$ contraction operator $T = P_{{\mathcal M}^{\perp}} S|_{{\mathcal M}^{\perp}}$, and \smallskip \noindent (3) via the \textbf{functional-model realization formula} $\Theta(z) = D + z C (I - z A)^{-1} B$ where $\left[ \begin{smallmatrix} A & B \\ C & D \end{smallmatrix} \right] = \left[ \begin{smallmatrix} S^{*} & S^{*} M_{\Theta} \\ \mathbf{ev}_{0} & \Theta(0) \end{smallmatrix} \right] \colon \left[ \begin{smallmatrix} {\mathcal M}^{\perp} \\ {\mathcal U} \end{smallmatrix} \right] \to \left[ \begin{smallmatrix} {\mathcal M}^{\perp} \\ {\mathcal Y} \end{smallmatrix} \right]$ with $\mathbf{ev}_{0} \colon f \mapsto f(0)$ is unitary. \smallskip \noindent We discuss analogues of these results for the weighted Bergman-shift operator $S_{n} \colon f(z) \mapsto z f(z)$ acting on the weighted Bergman space ${\mathcal A}_{n, {\mathcal Y}} = \{f(z) = \sum_{j \ge 0} f_{j} z^{j} :$ $\|f\|^{2} = \sum_{j \ge 0} \mu_{n,j} \| f_{j} \|^{2}_{\mathcal Y} \u3c \infty\}$ (where $\mu_{n,j} = 1/\left( \begin{smallmatrix} j+n-1 \\ j \end{smallmatrix} \right)$ are reciprocal binomial coefficients), or ${\mathcal A}_{n, {\mathcal Y}}$ is the Reproducing Kernel Hilbert Space with reproducing kernel equal to $K_{n,{\mathcal Y}}(z,w) = (1 - z \overline{w})^{-n} I_{\mathcal Y}$, as well as for the freely noncommutative weighted Bergman shift-tuple ${\mathbf S}_{n} = (S_{n,1}, \dots, S_{n,d})$ where $S_{n,j} \colon f(z) \mapsto f(z) z_{j}$ ($j = 1, \dots, d$) on the weighted Bergman-Fock space ${\mathcal A}_{n,{\mathcal Y}}({\mathbb F}_{d})$ consisting of formal power series $f(z) = \sum_{{\mathfrak a} \in {\mathbb F}_{d}} f_{\mathfrak a} z^{\mathfrak a}$ for which $\| f \|^{2} = \sum_{{\mathfrak a} \in {\mathbb F}_{d}} \mu_{n,|{\mathfrak a}|} \| f_{\mathfrak a} \|^{2} \u3c \infty$ (${\mathbb F}_{d}$ is the free semigroup (also called the monoid) on $d$ generators $1, \dots, d$, $z = (z_1, \dots, z_d)$ is a $d$-tuple of freely noncommuting indeterminates, $z^{\mathfrak a}$ is the noncommutative monomial $z^{\mathfrak a} = z_{i_{1}} \cdots z_{i_{N}}$ if ${\mathfrak a} = i_{1} \cdots i_{N}$, $|{\mathfrak a}|$ is the \textbf{length} of ${\mathfrak a}$ (equal to $N$ if ${\mathfrak a} = i_{1} \cdots i_{N}$). This talk reports on joint work with Vladimir Bolonikov of the College of William and Mary