Characterization of Projections

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


Proof


Sources