On the decomposable numerical range of operators

Abstract

‎Let $V$ be an $n$-dimensional complex inner product space‎. ‎Suppose‎ ‎$H$ is a subgroup of the symmetric group of degree $m$‎, ‎and‎ ‎$chi‎ :‎Hrightarrow mathbb{C} $ is an irreducible character (not‎ ‎necessarily linear)‎. ‎Denote by $V_{chi}(H)$ the symmetry class‎ ‎of tensors associated with $H$ and $chi$‎. ‎Let $K(T)in‎ (V_{chi}(H))$ be the operator induced by $Tin‎ ‎text{End}(V)$‎. ‎The decomposable numerical range $W_{chi}(T)$ of‎ ‎$T$ is a subset of the classical numerical range‎ ‎$W(K(T))$ of $K(T)$ defined as‎:‎$$‎ ‎W_{chi}(T)={(K(T)x^{ast }‎, ‎x^{ast}):x^{ast } is a‎ ‎decomposable unit tensor}‎.‎$$‎ ‎In this paper‎, ‎we study the interplay between the geometric‎ ‎properties of $W_{chi}(T)$ and the algebraic properties of $T$‎. ‎In fact‎, ‎we extend some of the results of ‎[‎‎C‎. ‎K‎. ‎Li and A‎. ‎Zaharia‎, ‎Decomposable numerical range on‎ ‎orthonormal decomposable tensors‎, Linear Algebra Appl. 308 ‎(2000), no, 1-3, 139--152] ‎and ‎[‎‎C‎. ‎K‎. ‎Li and A‎. ‎Zaharia‎, ‎Induced operators on symmetry classes‎ ‎of tensors‎, ‎Trans‎. ‎Amer‎. ‎Math‎. ‎Soc. 354 (2002), no. 2, 807--836]‎, ‎to non-linear irreducible characters‎.