Decomposable numerical ranges on orthonormal tensors Original Research Article

Let 1less-than-or-equals, slantmless-than-or-equals, slantn, and let image be a degree 1 character on a subgroup H of the symmetric group of degree m. The generalized matrix function on an m×m matrix B=(bij) associated with χ is defined by dχ(B)=∑σset membership, variantHχ(σ)∏j=1mbj,σ(j), and the decomposable numerical range of an n×n matrix A on orthonormal tensors associated with χ is defined byimageWχperpendicular(A)={dχ(X*AX):X isan n×mmatrixsuchthat X*X=Im}.We study relations between the geometrical properties of Wχperpendicular(A) and the algebraic properties of A, and determine the structure of those linear operators L on n×n complex matrices that satisfy Wχperpendicular(L(A))=Wχperpendicular(A) for all n×n matrices A. These results extend those of other researchers who treat the special cases of χ such as the principal or alternate character.