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 $\|\cdot\|_{\mathrm{bfd}}$ on the space $\mathrm{Hol}(E)$ of holomorphic functions on $E$ is defined by $$ \|f\|_{\mathrm{bfd}} \stackrel{\rm def}{=} \sup_{T\in \mathcal{F}_{\mathrm {bfd}}(E)}\|f(T)\|, $$ where $\mathcal{F}_{\mathrm {bfd}}(E)$ 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 $\|\cdot\|_{\mathrm{bfd}}$ is equivalent to the supremum norm $\|\cdot\|_\infty$ on $\mathrm{Hol}(E)$. 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 $\mathrm{Hol}(E)$ and the supremum norm $\|\cdot\|_\infty$ on the space $\mathrm{H}^\infty(G)$ of bounded holomorphic functions on the symmetrized bidisc, the domain $G$ in $\mathbb{C}^2$ defined by \begin{align*} G & \stackrel{\rm def}{=} \{(z+w,zw): |z|<1, |w|<1\}. \end{align*} It transpires that there exists a holomorphic embedding $\tau:E \to G$ having the property that, for any bounded holomorphic function $f$ on $E$, \[ \|f\|_{\mathrm{bfd}} = \inf\{\|F\|_\infty: F \in {\mathrm H}^\infty(G), F\circ\tau=f\}, \] and moreover, the infimum is attained at some $F \in \mathrm{H}^\infty(G)$. 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 $E$.