Characterization of Projections

Theorem
Let $\HH$ be a Hilbert space.

Let $A \in \map B \HH$ be an idempotent operator.

Then the following are equivalent:


 * $(1): \quad A$ is a projection
 * $(2): \quad A$ is the orthogonal projection onto $\Rng A$
 * $(3): \quad \norm A = 1$, where $\norm {\, \cdot \,}$ is the norm on bounded linear operators.
 * $(4): \quad A$ is Hermitian
 * $(5): \quad A$ is normal
 * $(6): \quad \forall h \in \HH: \innerprod {A h} h_\HH \ge 0$