Location
Cupples I Room 113
Start Date
7-18-2016 3:00 PM
End Date
18-7-2016 3:20 PM
Description
The band method plays a fundamental role in solving a Toeplitz and Nehari interpolation problem; see \cite{ggk2}. The solution to the Nehari problem involves the inverses of $I - HH^*$ and $I- H^*H$ where $H$ is the corresponding Hankel matrix. Here we will derive a similar result for a certain commutant lifing problem. Let $\Theta$ be an inner function in $H^{\infty}(\mathcal{E},\mathcal{Y})$ and $\mathcal{H}(\Theta)$ the subspace of $\ell_+^2(\mathcal{Y})$ defined by \[ \mathcal{H}(\Theta) = \ell_+^2(\mathcal{Y}) \ominus T_\Theta \ell_+^2(\mathcal{E}) \] where $T_\Theta$ is the Toeplitz operator determined by $\Theta$. Clearly, $\mathcal{H}(\Theta)$ is an invariant subspace for the backward shift $S_\mathcal{Y}^*$. Consider the \emph{data set} $\{A,T^\prime,S_\mathcal{Y} \}$ where $A$ is a strict contraction mapping $\ell_+^2(\mathcal{U)}$ into $\mathcal{H}(\Theta)$, the operator $T^\prime$ on $\mathcal{H}(\Theta)$ is the compression of $S_\mathcal{Y}$ to $\mathcal{H}(\Theta)$, that is, \[ T^\prime = \Pi_{\scriptscriptstyle \mathcal{H}(\Theta)} S_\mathcal{Y}| \mathcal{H}(\Theta) \mbox{ on } \mathcal{H}(\Theta). \] Here $\Pi_{\scriptscriptstyle \mathcal{H}(\Theta)}$ is the orthogonal projection from $\ell_+^2(\mathcal{Y})$ onto $\mathcal{H}(\Theta)$. Moreover, $A$ intertwines $S_\mathcal{U}$ with $T^\prime$, that is, $T^\prime A =AS_\mathcal{U}$. Given this data set the commutant lifting problem is to find all contractive Toeplitz operators $T_\Psi$ such that \begin{equation}\label{rclt} \Pi_{\scriptscriptstyle \mathcal{H}(\Theta)}T_\Psi =A. \end{equation} This lifting problem includes the Nevanlinna-Pick and Leech interpolation problems. Using two different methods we will show that the set of all solutions are given by \begin{align*} \Psi &= \big(\Upsilon_{12} + \Upsilon_{11} g\big) \big(\Upsilon_{22} + \Upsilon_{21} g\big)^{-1}. \end{align*} Here $g$ is a contactive analytic function acting between the appropriate spaces. Analogous to the band formulas in the Nehari interpolation problem, $\Upsilon_{jk}$ are determined by the inverses of $I-AA^*$ and $I- A^*A$. The proofs relay on different techniques. Finally, this is joint work with S. ter Horst and M.A. Kaashoek. %\end{abstract} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{thebibliography}{99} \bibitem{FFGK} C. Foias, A.E. Frazho, I. Gohberg, and M. A. Kaashoek, {\em Metric Constrained Interpolation, Commutant Lifting and Systems,} Operator Theory: Advances and Applications, {\bf 100}, Birkh\"{a}user-Verlag, 1998. \bibitem{ggk2} I. Gohberg, S. Goldberg, and M.A. Kaashoek, {\em Classes of Linear Operators}, Vol. II, Operator Theory: Advances and Applications, {\bf 63}, Birkh\"{a}user-Verlag, Basel, 1993. \end{thebibliography}
A band formula for a Toeplitz commutant lifting problem
Cupples I Room 113
The band method plays a fundamental role in solving a Toeplitz and Nehari interpolation problem; see \cite{ggk2}. The solution to the Nehari problem involves the inverses of $I - HH^*$ and $I- H^*H$ where $H$ is the corresponding Hankel matrix. Here we will derive a similar result for a certain commutant lifing problem. Let $\Theta$ be an inner function in $H^{\infty}(\mathcal{E},\mathcal{Y})$ and $\mathcal{H}(\Theta)$ the subspace of $\ell_+^2(\mathcal{Y})$ defined by \[ \mathcal{H}(\Theta) = \ell_+^2(\mathcal{Y}) \ominus T_\Theta \ell_+^2(\mathcal{E}) \] where $T_\Theta$ is the Toeplitz operator determined by $\Theta$. Clearly, $\mathcal{H}(\Theta)$ is an invariant subspace for the backward shift $S_\mathcal{Y}^*$. Consider the \emph{data set} $\{A,T^\prime,S_\mathcal{Y} \}$ where $A$ is a strict contraction mapping $\ell_+^2(\mathcal{U)}$ into $\mathcal{H}(\Theta)$, the operator $T^\prime$ on $\mathcal{H}(\Theta)$ is the compression of $S_\mathcal{Y}$ to $\mathcal{H}(\Theta)$, that is, \[ T^\prime = \Pi_{\scriptscriptstyle \mathcal{H}(\Theta)} S_\mathcal{Y}| \mathcal{H}(\Theta) \mbox{ on } \mathcal{H}(\Theta). \] Here $\Pi_{\scriptscriptstyle \mathcal{H}(\Theta)}$ is the orthogonal projection from $\ell_+^2(\mathcal{Y})$ onto $\mathcal{H}(\Theta)$. Moreover, $A$ intertwines $S_\mathcal{U}$ with $T^\prime$, that is, $T^\prime A =AS_\mathcal{U}$. Given this data set the commutant lifting problem is to find all contractive Toeplitz operators $T_\Psi$ such that \begin{equation}\label{rclt} \Pi_{\scriptscriptstyle \mathcal{H}(\Theta)}T_\Psi =A. \end{equation} This lifting problem includes the Nevanlinna-Pick and Leech interpolation problems. Using two different methods we will show that the set of all solutions are given by \begin{align*} \Psi &= \big(\Upsilon_{12} + \Upsilon_{11} g\big) \big(\Upsilon_{22} + \Upsilon_{21} g\big)^{-1}. \end{align*} Here $g$ is a contactive analytic function acting between the appropriate spaces. Analogous to the band formulas in the Nehari interpolation problem, $\Upsilon_{jk}$ are determined by the inverses of $I-AA^*$ and $I- A^*A$. The proofs relay on different techniques. Finally, this is joint work with S. ter Horst and M.A. Kaashoek. %\end{abstract} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{thebibliography}{99} \bibitem{FFGK} C. Foias, A.E. Frazho, I. Gohberg, and M. A. Kaashoek, {\em Metric Constrained Interpolation, Commutant Lifting and Systems,} Operator Theory: Advances and Applications, {\bf 100}, Birkh\"{a}user-Verlag, 1998. \bibitem{ggk2} I. Gohberg, S. Goldberg, and M.A. Kaashoek, {\em Classes of Linear Operators}, Vol. II, Operator Theory: Advances and Applications, {\bf 63}, Birkh\"{a}user-Verlag, Basel, 1993. \end{thebibliography}