Krein Space Numerical Ranges: Compressions and Dilations

Ela Krein Spaces Numerical Ranges and Their Computer Generation

Let J be an involutive Hermitian matrix with signature (t, n− t), 0 ≤ t ≤ n, that is, with t positive and n− t negative eigenvalues. The Krein space numerical range of a complex matrix A of size n is the collection of complex numbers of the form ξ ∗JAξ ξ∗Jξ , with ξ ∈ Cn and ξ∗Jξ = 0. In this note, a class of tridiagonal matrices with hyperbolical numerical range is investigated. A Matlab progr...

Constraint Unitary Dilations and Numerical Ranges

It is shown that each contraction A on a Hilbert space H, with A + A I for some 2 R, has a unitary dilation U on H H satisfying U + U I. This is used to settle a conjecture of Halmos in the aarmative: The closure of the numerical range of each contraction A is the intersection of the closures of the numerical ranges of all unitary dilations of A. By means of the duality theory of completely pos...

Shapes and computer generation of numerical ranges of Krein space operators

The numerical range of linear operators on the 2-dimensional Krein space

The aim of this note is to provide the complete characterization of the numerical range of linear operators on the 2-dimensional Krein space C.

Generalized Scalar Operators as Dilations

It is shown that polynomially bounded operators on Banach spaces have polynomially bounded dilations which have spectrum in the unit circle and are generalized scalar. The proof also yields a description of all compressions of generalized scalar operators with spectrum in the unit circle.