Teixeira, Ricardo Emanuel CunhaBebiano, NatáliaProvidência, João da2012-05-042012-05-042007-07Teixeira, Ricardo; Bebiano, Natália; Providência, João da. The Elliptical and the Hyperbolical Range Theorems Revisited. In "Proceedings of the International Conference on Engineering and Mathematics (ENMA 2007), Bilbao, 2007".http://hdl.handle.net/10400.3/1340International Conference on Engineering and Mathematics (ENMA'2007), July 9-11, 2007 - Bilbao, Spain.A geometrical proof of the Hyperbolical Range Theorem, concerning the numerical range of linear operators on 2-dimensional Krein spaces, is given. The classical Elliptical Range Theorem, which is the correspondent result for Hilbert spaces, is also obtained using the same technique. Both proofs depend on the geometrical properties of the plane algebraic curve that generates the numerical range as its convex hull or as its pseudoconvex hull.engGeneralized Numerical RangeIndefinite Inner Product SpacePlane Algebraic CurveNumerical Rangeconference object2012-04-29