The pseudospectra of linear combinations of two orthogonal projections in the Hilbert space

Abstract

Let $P$ and $Q$ be two orthogonal projections in Hilbert space $\mathcal{H}$. For $\alpha, \beta \in \mathbb{C}\backslash \{0\}$, the lower bound and the upper bound of the pseudospectra of $\alpha P+ \beta Q$ are obtained. The bounds are represented by the product $P Q$ which are independent of the choice of scalars $\alpha,\beta$.
For $\alpha,\beta \in \mathbb{C}\backslash \{0\},\alpha+\beta \neq \xi,\xi \in \mathbb{C}$, the bounds of the pseudospectra of $\alpha P +\beta Q -\xi PQ $ are also obtained in the same way. Finally, two examples are constructed to show the effectiveness of the results.