On the boundedness of quaternionic numerical ranges with respect to nonstandard involutions

Let ϕ be a nonstandard involution on the set of all quaternion numbers, and α be a quaternion such that ϕ(α)=α. For any n×n quaternion matrix A, the notion of numerical range of A with respect to ϕ, Wϕ(α)(A), was introduced by L. Rodman in 2014, and for the case α=0, he characterized all quaternion matrices A whose Wϕ(0)(A) is bounded. In this paper, for the case α≠0, those quaternion matrices A for which Wϕ(α)(A) is bounded, are characterized. Also, by introducing the notion of ϕ-class of a quaternion number, all quaternion matrices A whose Wϕ(α)(A) is a singleton set, are characterized, and for n≥3, it is shown that Wϕ(α)(A) is bounded if and only if it is a singleton set.