The complex geomety of a domain related to $\mu$-synthesis

Abstract

We describe the basic complex geometry and function theory of the {\em pentablock} $\mathcal{P}$, which is the bounded domain in ${\mathbb{C}}^3$ given by \[ \mathcal{P}= \{(a_{21}, \mathrm{tr} A, \det A): A= \begin{bmatrix} a_{ij}\end{bmatrix}_{i,j=1}^2 \in \mathbb{B}\} \] where $\mathbb{B}$ denotes the open unit ball in the space of $2\times 2$ complex matrices. We prove several characterizations of the domain. We describe its distinguished boundary and exhibit a $4$-parameter group of automorphisms of $\mathcal{P}$. We show that $\mathcal{P}$ is intimately connected with the problem of $\mu$-synthesis for a certain cost function $\mu$ on the space of $2\times 2$ matrices defined in connection with robust stabilization by control engineers. We demonstrate connections between the function theories of $\mathcal{P}$ and $\mathbb{B}$. We show that $\mathcal{P}$ is polynomially convex and starlike.

36 pages, 2 figures. This version contains corrections of some inaccuracies and an expanded argument for Proposition 12.4

References in corpus (3)

• Operator theory on the tetrablock
• Some analysable instances of mu-synthesis
• The group of automorphisms of the pentablock

Cited by in corpus (5)

• Rigidity of Proper Holomorphic Self-mappings of the Pentablock
• Geometric properties of domains related to $\mu$-synthesis
• Bounded Real Lemma and structured singular value versus diagonal scaling: the free noncommutative setting
• The group of automorphisms of the pentablock
• Operator Theory on the Pentablock