On the operators with numerical range in an ellipse

We give new necessary and sufficient conditi+ons for the numerical range W(T) of an operator T is an element of B(H) to be a subset of the closed elliptical set K-delta (def)(=) {d + iy: x(2)/(1+delta(2) + y2 /(1-delta)-<= 1} where 0 < delta < 1. Here B(H) denotes the collection of bounded linear operators on a Hilbert space 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. Specifically, we show that, if T acts on a finite-dimensional Hilbert space H and satisfies a certain genericity assumption, then W(T) C Ka if and only if there exists a Hilbert space KH, operators X1 and X2 on H and a unitary operator U acting on K such that and X-1+X-2=T, X-1 X-2=delta (0.1) and X + X = 2PHU* |(H), k = 1, 2,..., (0.2) where P-H denotes the orthogonal projection from K to H. We next generalize the lemma of Sarason that describes power dilations in terms of semi-invariant subspaces to operators T that satisfy the relations (0.1) and (0.2). This generalization yields a characterization of the operators T is an element of B(H) such that W(T) is contained in Ks in terms of certain structured contractions that act on HH. 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) <= Ks. We prove that, if T acts on a finite- dimensional Hilbert space H, then W(T) C K if and only if there exist a pair of contractions A, B is an element of B(H) such that A is self-adjoint and T = 2 root 8 A+ (1 - 8) root 1 + A B root 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-extension problem for analytic functions defined on a slice of the symmetrized bidisc. (c) 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).