a b s t r a c tThe numerical range of an operator is a well studied concept with many applications in several areas of mathematics. In this paper, the numerical range of banded biperiodic Toeplitz operators is investigated, performing a reduction to the 2 × 2 case. Namely, the parametric equations of the boundary generating curves are deduced and two algorithms for the numerical generation of the numerical range are presented.
Abstract. 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, with ξ ∈ C n and ξ * Jξ = 0. In this note, a class of tridiagonal matrices with hyperbolical numerical range is investigated. A Matlab program is developed to generate Krein spaces numerical ranges in the finite dimensional case.
We propose efficient methods for the numerical approximation of the field of values of the linear pencil A − λ B, when one of the matrix coefficients A or B is Hermitian and λ ∈ . Our approach builds on the fact that the field of values can be reduced under compressions to the bidimensional case, for which these sets can be exactly determined. The presented algorithms hold for matrices both of small and large size. Furthermore, we investigate spectral inclusion regions for the pencil based on certain fields of values. The results are illustrated by numerical examples. We point out that the given procedures complement the known ones in the literature.
Consider the Hilbert space (H; h ; i) equipped with the indefinite inner product OEu; v D v J u, u; v 2 H, where J is an indefinite self-adjoint involution acting on H. The Krein space numerical range W J .T / of an operator T acting on H is the set of all the values attained by the quadratic form OET u; u, with u 2 H satisfying OEu; u D˙1. We develop, implement and test an alternative algorithm to compute W J .T / in the finite dimensional case, constructing 2 by 2 matrix compressions of T and their easily determined elliptical and hyperbolical numerical ranges. The numerical results reported here indicate that this method is very efficient, since it is faster and more accurate than either of the existing algorithms. Further, it may yield easy solutions for the inverse indefinite numerical range problem. Our algorithm uses an idea of Marcus and Pesce from 1987 for generating Hilbert space numerical ranges of matrices of size n.
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