Function theory in the bfd-norm on an elliptical region

Abstract

Let E be the open region in the complex plane bounded by an ellipse. The B. and F. Delyon norm ‖⋅‖bfd on the space Hol(E) of holomorphic functions on E is defined by‖f‖bfd=defsupT∈Fbfd(E)⁡‖f(T)‖, where Fbfd(E) is the class of operators T such that the closure of the numerical range of T is contained in E. The name of the norm recognizes a celebrated theorem of the brothers Delyon, which implies that ‖⋅‖bfd is equivalent to the supremum norm ‖⋅‖∞ on Hol(E).The purpose of this paper is to develop the theory of holomorphic functions of bfd-norm less than or equal to one on E. To do so we shall employ a remarkable connection between the bfd norm on Hol(E) and the supremum norm ‖⋅‖∞ on the space H∞(G) of bounded holomorphic functions on the symmetrized bidisc, the domain G in C2 defined byG=def{(z+w,zw):|z|<1,|w|<1}. It transpires that there exists a holomorphic embedding τ:E→G having the property that, for any bounded holomorphic function f on E,‖f‖bfd=inf⁡{‖F‖∞:F∈H∞(G),F∘τ=f}, and moreover, the infimum is attained at some F∈H∞(G). This result allows us to derive, for holomorphic functions of bfd-norm at most one on E, analogs of the well-known model and realization formulae for Schur-class functions. We also give a second derivation of these models and realizations, which exploits the Zhukovskii mapping from an annulus onto E.