Characterization of Projections

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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 operator norm
$(4): \qquad A$ is self-adjoint
$(5): \qquad A$ is normal
$(6): \qquad \forall h \in H: \left\langle{Ah, h}\right\rangle_H \ge 0$


Proof


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