Characterization of Projections

Theorem

Let $H$ be a Hilbert space.

Let $A \in B \left({H}\right)$ be an idempotent operator.

Then the following are equivalent:

$(1): \qquad A$ is a projection

$(2): \qquad A$ is the orthogonal projection onto $\operatorname{ran} A$

$(3): \qquad \left\Vert{A}\right\Vert = 1$, where $\left\Vert{\cdot}\right\Vert$ is the norm on bounded linear operators.

$(4): \qquad A$ is self-adjoint

$(5): \qquad A$ is normal

$(6): \qquad \forall h \in H: \left\langle{Ah, h}\right\rangle_H \ge 0$

Sources

• 1990: John B. Conway: A Course in Functional Analysis ... (previous) ... (next) $II.3.3$