24,544 research outputs found
The complex geomety of a domain related to -synthesis
We describe the basic complex geometry and function theory of the {\em
pentablock} , which is the bounded domain in given
by where denotes
the open unit ball in the space of complex matrices. We prove
several characterizations of the domain. We describe its distinguished boundary
and exhibit a -parameter group of automorphisms of . We show
that is intimately connected with the problem of -synthesis
for a certain cost function on the space of matrices defined
in connection with robust stabilization by control engineers. We demonstrate
connections between the function theories of and . We
show that is polynomially convex and starlike.Comment: 36 pages, 2 figures. This version contains corrections of some
inaccuracies and an expanded argument for Proposition 12.
A geometric characterization of the symmetrized bidisc
The symmetrized bidisc
G def = {(z + w, zw) : |z| < 1, |w| < 1}
has interesting geometric properties. While it has a plentiful supply of complex geodesics and of automorphisms, there is nevertheless a unique complex geodesic in G that is invariant under all automorphisms of G. Moreover, G is foliated by those complex geodesics that meet in one point and have nontrivial stabilizer. We prove that these properties, together with two further geometric hypotheses on the action of the automorphism group of G, characterize the symmetrized bidisc in the class of complex manifolds
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