Location
Cupples I Room 207
Start Date
7-18-2016 3:00 PM
End Date
18-7-2016 3:20 PM
Description
\noindent We give a solvability criterion for the following $\mu$-synthesis problem. Let $\mu$ be the structured singular value for the diagonal matrices with entries in $\Bbb{C}$. \vspace{0.2cm} \noindent\text{\bf{Problem.}} Given distinct points $\lambda_1,\dots,\lambda_n$ in the open unit disc $\Bbb{D}$ and target $2\times2$ complex matrices $W_1, \dots, W_n$ such that $\mu(W_j)\leq 1$ for all $j=1,\dots, n$, find a holomorphic $2\times2$ matrix function $F$ on $\Bbb{D}$ such that $F(\lambda_j)=W_j$ for each $j$, and $\mu(F(\lambda))\leq1$ for all $\lambda\in\Bbb{D}$. \vspace{0.2cm} \noindent By [1, Theorem 9.2], this problem is equivalent to the following interpolation problem: does there exist a holomorphic function $x$ from the disc to the tetrablock $\overline{\Bbb{E}}$ such that $x(\lambda_j)=(w_{11}^j,w_{22}^j,\det{W_j})$ for each $j$? The tetrablock is the domain in $\Bbb{C}^3$ defined by \[\overline{\Bbb{E}}:=\{(x_1,x_2,x_3)\in\Bbb{C}^3:1-x_1z-x_2w+x_3zw\neq0\text{ for all }z,w\in\Bbb{D}\}.\] \vspace{0.2cm} \noindent In this talk we show such an $x$ exists if and only if, for distinct $z_1,z_2,z_3\in\Bbb{D}$, there are positive $3n$-square matrices $[N_{il,jk}]$, of rank $1$, and $[M_{il,jk}]$ such that \[\left[1-\overline{\frac{z_lx_{3i}-x_{1i}}{x_{2i}z_l-1}}\frac{z_kx_{3j}-x_{1j}}{x_{2j}z_k-1}\right] \geq[(1-\overline{z_l}z_k)N_{il,jk}]+[(1-\overline{\lambda_i}\lambda_j)M_{il,jk}],\] where $(x_{1j},x_{2j},x_{3j})=(w_{11}^j,w_{22}^j,\det{W_j})$ for each $j$.\\ \noindent The talk is based on a joint work with Z. A. Lykova and N. J. Young.\\ \noindent [1] A. A. Abouhajar, M. C. White and N. J. Young, A Schwarz lemma for a domain related to $\mu$-synthesis, \emph{J. Geom. Anal. 17}, (2007), pp. 717-750
A criterion for the solvability of a $\mu$-synthesis problem
Cupples I Room 207
\noindent We give a solvability criterion for the following $\mu$-synthesis problem. Let $\mu$ be the structured singular value for the diagonal matrices with entries in $\Bbb{C}$. \vspace{0.2cm} \noindent\text{\bf{Problem.}} Given distinct points $\lambda_1,\dots,\lambda_n$ in the open unit disc $\Bbb{D}$ and target $2\times2$ complex matrices $W_1, \dots, W_n$ such that $\mu(W_j)\leq 1$ for all $j=1,\dots, n$, find a holomorphic $2\times2$ matrix function $F$ on $\Bbb{D}$ such that $F(\lambda_j)=W_j$ for each $j$, and $\mu(F(\lambda))\leq1$ for all $\lambda\in\Bbb{D}$. \vspace{0.2cm} \noindent By [1, Theorem 9.2], this problem is equivalent to the following interpolation problem: does there exist a holomorphic function $x$ from the disc to the tetrablock $\overline{\Bbb{E}}$ such that $x(\lambda_j)=(w_{11}^j,w_{22}^j,\det{W_j})$ for each $j$? The tetrablock is the domain in $\Bbb{C}^3$ defined by \[\overline{\Bbb{E}}:=\{(x_1,x_2,x_3)\in\Bbb{C}^3:1-x_1z-x_2w+x_3zw\neq0\text{ for all }z,w\in\Bbb{D}\}.\] \vspace{0.2cm} \noindent In this talk we show such an $x$ exists if and only if, for distinct $z_1,z_2,z_3\in\Bbb{D}$, there are positive $3n$-square matrices $[N_{il,jk}]$, of rank $1$, and $[M_{il,jk}]$ such that \[\left[1-\overline{\frac{z_lx_{3i}-x_{1i}}{x_{2i}z_l-1}}\frac{z_kx_{3j}-x_{1j}}{x_{2j}z_k-1}\right] \geq[(1-\overline{z_l}z_k)N_{il,jk}]+[(1-\overline{\lambda_i}\lambda_j)M_{il,jk}],\] where $(x_{1j},x_{2j},x_{3j})=(w_{11}^j,w_{22}^j,\det{W_j})$ for each $j$.\\ \noindent The talk is based on a joint work with Z. A. Lykova and N. J. Young.\\ \noindent [1] A. A. Abouhajar, M. C. White and N. J. Young, A Schwarz lemma for a domain related to $\mu$-synthesis, \emph{J. Geom. Anal. 17}, (2007), pp. 717-750