Self Adjoint operator $\Rightarrow$ Idempotent Operator?

Question

If $P\in\mathcal{L}(H,H)$, with $H$ a Hilbert space, such that $P = P^*$, Is possible to show that $P^2 = P$?

If that is possible, then $P$ is a projection operator, right?

Thanks in advance.

Answer 1

An operator $P$ satisfying $P = P^{\ast}$ is called self-adjoint. There are plenty of self-adjoint operators that do not satisfy $P^2 = P$; for example, $P = 2I$, as proximal mentioned.

An operator $P$ satisfying $P = P^2$ is called a projection. Not all projections are self-adjoint, such as $\begin{bmatrix} 1 & 1 \\ 0 & 0 \end{bmatrix}$.

An operator that satisfies both is called an orthogonal projection.