Abstract

We give new necessary and sufficient conditions for the numerical range W(T) of an operator T∈B(H) to be a subset of the closed elliptical set Kδ⊆C given byKδ=def{x+iy:x2(1+δ)2+y2(1−δ)2≤1}, where 0<δ<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)⊆Kδ if and only if there exists a Hilbert space K⊇H, operators X1 and X2 on H and a unitary operator U acting on K such that(0.1)X1+X2=T,X1X2=δ and(0.2)X1k+X2k=2PHUk|H,k=1,2,…, where PH 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∈B(H) such that W(T) is contained in Kδ in terms of certain structured contractions that act on H⊕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)⊆Kδ. We prove that, if T acts on a finite-dimensional Hilbert space H, then W(T)⊆Kδ if and only if there exist a pair of contractions A,B∈B(H) such that A is self-adjoint andT=2δA+(1−δ)1+AB1−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.

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call

Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.