On the Operators with Numerical Range in an Ellipse

Abstract

We give new necessary and sufficient conditions for the numerical range $W(T)$ of an operator $T \in \mathcal{B}(\mathcal{H})$ to be a subset of the closed elliptical set $K_δ\subseteq \mathbb{C}$ given by \[ K_δ{\stackrel{\rm def}{=}} \left\{x+iy: \frac{x^2}{(1+δ)^2} + \frac{y^2}{(1-δ)^2} \leq 1\right\}, \] where $0 < δ< 1$. Here $\mathcal{B}(\mathcal{H})$ denotes the collection of bounded linear operators on a Hilbert space $\mathcal{H}$. Central to our efforts is a direct generalization of Berger's well-known criterion for an operator to have numerical radius at most one, his so-called strange dilation theorem. We next generalize the lemma of Sarason that describes power dilations in terms of semi-invariant subspaces to operators $T$ that satisfy appropriate dilation properties. This generalization yields a characterization of the operators $T\in \mathcal{B}(\mathcal{H})$ such that $W(T)$ is contained in $K_δ$ in terms of certain structured contractions that act on $\mathcal{H} \oplus \mathcal{H}$. As a corollary of our results we extend Ando's parametrization of operators having numerical range in a disc to those $T$ such that $W(T)\subseteq K_δ$. We prove that, if $T$ acts on a finite-dimensional Hilbert space $\mathcal{H}$, then $W(T)\subseteq K_δ$ if and only if there exist a pair of contractions $A,B \in \mathcal{B}(\mathcal{H})$ such that $A$ is self-adjoint and \[ T=2\sqrtδA + (1-δ)\sqrt{1+A}\ B\sqrt{1-A}. \] We also obtain a formula for the B. and F. Delyon calcular norm of an analytic function on the inside of an ellipse in terms of the extremal $H^\infty$-extension problem for analytic functions defined on a slice of the symmetrized bidisc.