Characterization of Projections

Theorem

Let $H$ be a Hilbert space.

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

Then the following are equivalent:

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

$(2): \qquad A$ is the orthogonal projection onto $\Rng A$

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

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

$(5): \qquad A$ is normal

$(6): \qquad \forall h \in H: \innerprod {A h} h_H \ge 0$

Sources

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