Krein space numerical ranges: compressions and dilations

Krein Space Numerical Ranges: Compressions and Dilations

A criterion for the numerical range of a linear operator acting in a Krein space to be a two-component hyperbolical disc is given, using the concept of support function. A characterization of the Krein space numerical range as a union of hyperbolical discs is obtained by a reduction to the two-dimensional case. We revisit a famous result of Ando concerning the inclusion relation W (A) ⊆ W (B) o...

Higher-rank Numerical Ranges and Dilations

For any n-by-n complex matrix A and any k, 1 ≤ k ≤ n, let Λk(A) = {λ ∈ C : X∗AX = λIk for some n-by-k X satisfying X∗X = Ik} be its rank-k numerical range. It is shown that if A is an n-by-n contraction, then Λk(A) = ∩{Λk(U) : U is an (n + dA)-by-(n + dA) unitary dilation of A}, where dA = rank (In − A∗A). This extends and refines previous results of Choi and Li on constrained unitary dilations...

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...

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...

Shapes and computer generation of numerical ranges of Krein space operators