On quaternionic numerical ranges with respect to nonstandard involutions

Let ϕ be a nonstandard involution on the set of all quaternions, and the quaternion α be such that ϕ(α)=α. The notion of numerical range of an n×n quaternion matrix A with respect to ϕ was introduced by Leiba Rodman (2014) [8] asWϕ(α)(A)={xϕAx:x is an n×1 quaternion vector andxϕx=α}, where for x=[x1⋯xn]T, xϕ=[ϕ(x1)⋯ϕ(xn)]. In this paper, some algebraic and geometrical properties of Wϕ(0)(.) for every arbitrary quaternion matrix are investigated. Moreover, a description of this set is given for 2×2 quaternion matrices, and Wϕ(0)(.) is characterized for ϕ-hermitian and ϕ-skewhermitian quaternion matrices. To illustrate the main results, some examples are also given.