Let P and Q be two orthogonal projections in Hilbert space H. For α, β ∈ C \0, the lower bound and the upper bound of the pseudospectra of αP + βQ are obtained. The bounds are represented by the product PQ which are independent of the choice of scalars α, β. For α, β ∈ C \0, α + β, ξ, ξ ∈ C, the bounds of the pseudospectra of αP + βQ-ξPQ are also obtained in the same way. Finally, two examples are constructed to show the effectiveness of the results.
Xue et al. (Wed,) studied this question.