The complex geometry of a domain related to μ-synthesis

We establish the basic complex geometry and function theory of the pentablock P, which is the bounded domainP={(a21,trA,det⁡A):A=[aij]i,j=12∈B} where B denotes the open unit ball in the space of 2×2 complex matrices. We prove several characterisations of the domain. We show that P arises naturally in connection with a certain robust stabilisation problem in control theory, the problem of μ-synthesis. We describe the distinguished boundary of P and exhibit a 4-parameter group of automorphisms of P. We demonstrate connections between the function theories of P and B. We show that P is polynomially convex and starlike, and we show that the real pentablock P∩R3 is a convex set bounded by five faces, three of them flat and two curved.