State space formulas for a suboptimal rational Leech problem I: Maximum entropy solution

For the strictly positive case (the suboptimal case) the maximum entropy solution $X$ to the Leech problem $G(z)X(z)=K(z)$ and $\|X\|_\infty=\sup_{|z|\leq 1}\|X(z)\|\leq 1$, with $G$ and $K$ stable rational matrix functions, is proved to be a stable rational matrix function. An explicit state space realization for $X$ is given, and $\|X\|_\infty$ turns out to be strictly less than one. The matrices involved in this realization are computed from the matrices appearing in a state space realization of the data functions $G$ and $K$. A formula for the entropy of $X$ is also given.Comment: 19 page

A band formula for a Toeplitz commutant lifting problem

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

Models for noncommuting operators

AbstractThis paper develops a model theory for a pair of noncommuting operators. Using backward shift operators on a Fock space Rota's Theorem is generalized, i.e., it is shown that any two bounded operators on a Hilbert space are simultaneously similar to part of a pair of backward shift operators on a Fock space. These shift operators and the Fock space framework are also used to develop a dilation theory for two noncommuting operators

State space formulas for stable rational matrix solutions of a Leech problem

Given stable rational matrix functions $G$ and $K$, a procedure is presented to compute a stable rational matrix solution $X$ to the Leech problem associated with $G$ and $K$, that is, $G(z)X(z)=K(z)$ and $\sup_{|z|\leq 1}\|X(z)\|\leq 1$. The solution is given in the form of a state space realization, where the matrices involved in this realization are computed from state space realizations of the data functions $G$ and $K$.Comment: 25 page

State space formulas for a suboptimal rational Leech problem II: Parametrization of all solutions

For the strictly positive case (the suboptimal case), given stable rational matrix functions $G$ and $K$, the set of all $H^\infty$ solutions $X$ to the Leech problem associated with $G$ and $K$, that is, $G(z)X(z)=K(z)$ and $\sup_{|z|\leq 1}\|X(z)\|\leq 1$, is presented as the range of a linear fractional representation of which the coefficients are presented in state space form. The matrices involved in the realizations are computed from state space realizations of the data functions $G$ and $K$. On the one hand the results are based on the commutant lifting theorem and on the other hand on stabilizing solutions of algebraic Riccati equations related to spectral factorizations.Comment: 28 page

All solutions to the relaxed commutant lifting problem

A new description is given of all solutions to the relaxed commutant lifting problem. The method of proof is also different from earlier ones, and uses only an operator-valued version of a classical lemma on harmonic majorants.Comment: 15 page