The absolute values and support projections for a class of operator matrices involving idempotents

Let $\lambda\in \mathbb{R},$ $\mu\in \mathbb{R}$ and $B$ be a linear bounded operator from a Hilbert space $\mathcal{K}$ into another Hilbert space $\mathcal{H}.$ In this paper, we consider the formulas of the absolute value $|Q_{\lambda,\mu}|,$ where $Q_{\lambda,\mu}$ with respect to the decomposition $\mathcal{H}\oplus\mathcal{K}$ have the operator matrix form $Q_{\lambda,\mu}:=\left(\begin{array}{cc}\lambda I&B\\B^*&\mu I\end{array}\right).$ Then the positive part and the support projection of $Q_{\lambda,0}$ are obtained. Also, we characterize the symmetry $J$ such that a projection $E$ is the $J$-projection. In particular, the minimal element of the set of all symmetries $J$ with the property $JE\geqslant0$ is described.