More on geometry of Krein space C-numerical range

Alexander Guterman,Rute Lemos,Graça Soares

doi:10.1016/j.amc.2019.01.029

Abstract

For n × n complex matrices A, C and H, where H is non-singular Hermitian, the Krein space C-numerical range of A induced by H is the subset of the complex plane given by {Tr(CU[*]AU):U−1=U[*]} with U[*]=H−1U*H the H-adjoint matrix of U. We revisit several results on the geometry of Krein space C-numerical range of A and in particular we obtain a condition for the Krein space C-numerical range to be a subset of the real line.

Highlights

Let Mn denote the algebra of n × n complex matrices, In denote the identity matrix of order n and H ∈ Mn be a non-singular Hermitian matrix
The main goal of this paper is to investigate the cases when the Krein space C-numerical range of A induced by J is either a singleton or a subset of the real line
We find a correct criteria by showing, when C ∈ Mn is a J-unitarily diagonalizable with real eigenvalues and A ∈ Mn, that WJC (A) is a real subset if and only if one of the following conditions holds: (a) A is J-Hermitian; (b) ImJ A is a non-zero scalar matrix and C has null trace; (c) C is a scalar matrix and A has real trace

Summary

Introduction

Let Mn denote the algebra of n × n complex matrices, In denote the identity matrix of order n and H ∈ Mn be a non-singular Hermitian matrix. The main goal of this paper is to investigate the cases when the Krein space C-numerical range of A induced by J is either a singleton or a subset of the real line. The first result due to Bebiano et al [1, Theorem 5.1] concerns the (J, C)tracial range, when C is a diagonal matrix and JC has pairwise distinct main diagonal entries, and it states that VJ,C (A) is a singleton if and only if JA is a scalar matrix. Assuming that C is a real diagonal matrix, JC has pairwise distinct main diagonal entries and A ∈ Mn, it is stated in [1, Theorem 5.2] that VJ,C (A) is a subset of the real line if and only if A is Hermitian.

Preliminary results on the Krein space numerical range

Preliminary results on the Krein space C-numerical range

Krein space C-numerical range as a singleton

Krein space C-numerical range as a subset of a line