Location
Crow 206
Start Date
7-19-2016 3:30 PM
End Date
19-7-2016 3:50 PM
Description
\documentclass[reqno]{amsart} \begin{document} \begin{center} {\Large{\bf Beurling-Lax type theorems in weighted Bergman-Fock spaces}} \end{center} \bigskip \begin{center} {\large{Vladimir Bolotnikov, The College of William and Mary}} \end{center} \bigskip Since the shift operator $M_z: \, f(z)\to zf(z)$ is an isometry on the $\mathcal Y$-valued Hardy space $H_{\mathcal Y}^{2}$ of the open unit disk, any $M_z$-invariant closed subspace $\mathcal M\subset H^2_{\mathcal Y}$ is generated by the wandering subspace $\mathcal E=\mathcal M\ominus z\mathcal M=P_{\mathcal M}z\mathcal M^\perp$. Furthermore, $z^k\mathcal E \perp z^\ell \mathcal E$ for $k\neq \ell$, and any wandering subspace has the form $\mathcal E=\Theta\mathcal U$ for some $\mathcal L(\mathcal U,\mathcal Y)$-valued inner function $\Theta$ and an appropriate coefficient space $\mathcal U$, which in turn leads to the representations $$ \mathcal M=\bigoplus_{k\ge 0}(z^k\mathcal M\ominus z^{k+1}\mathcal M)=\bigoplus_{k\ge 0}z^k\mathcal E=\bigoplus_{k\ge 0}z^k(P_{\mathcal M}z\mathcal M^\perp)=\bigoplus_{k\ge 0}z^k\Theta\mathcal U=\Theta H^2_{\mathcal U} $$ for an $M_z$-invariant subspace $\mathcal M\subset H^2_{\mathcal Y}$. These equivalent representations display the Beurling-Lax theorem and admit extensions to the noncommutative Fock space setting of formal power series in several non-commuting variables. We will discuss their possible extensions in the context of weighted Bergman-Fock spaces where they are produce several non-equivalent representations for closed subspaces invariant under multiplication by coordinate functions. \end{document}
Beurling-Lax type theorems in weighted Bergman-Fock spaces
Crow 206
\documentclass[reqno]{amsart} \begin{document} \begin{center} {\Large{\bf Beurling-Lax type theorems in weighted Bergman-Fock spaces}} \end{center} \bigskip \begin{center} {\large{Vladimir Bolotnikov, The College of William and Mary}} \end{center} \bigskip Since the shift operator $M_z: \, f(z)\to zf(z)$ is an isometry on the $\mathcal Y$-valued Hardy space $H_{\mathcal Y}^{2}$ of the open unit disk, any $M_z$-invariant closed subspace $\mathcal M\subset H^2_{\mathcal Y}$ is generated by the wandering subspace $\mathcal E=\mathcal M\ominus z\mathcal M=P_{\mathcal M}z\mathcal M^\perp$. Furthermore, $z^k\mathcal E \perp z^\ell \mathcal E$ for $k\neq \ell$, and any wandering subspace has the form $\mathcal E=\Theta\mathcal U$ for some $\mathcal L(\mathcal U,\mathcal Y)$-valued inner function $\Theta$ and an appropriate coefficient space $\mathcal U$, which in turn leads to the representations $$ \mathcal M=\bigoplus_{k\ge 0}(z^k\mathcal M\ominus z^{k+1}\mathcal M)=\bigoplus_{k\ge 0}z^k\mathcal E=\bigoplus_{k\ge 0}z^k(P_{\mathcal M}z\mathcal M^\perp)=\bigoplus_{k\ge 0}z^k\Theta\mathcal U=\Theta H^2_{\mathcal U} $$ for an $M_z$-invariant subspace $\mathcal M\subset H^2_{\mathcal Y}$. These equivalent representations display the Beurling-Lax theorem and admit extensions to the noncommutative Fock space setting of formal power series in several non-commuting variables. We will discuss their possible extensions in the context of weighted Bergman-Fock spaces where they are produce several non-equivalent representations for closed subspaces invariant under multiplication by coordinate functions. \end{document}