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

$(6): \quad \forall h \in \HH: \innerprod {A h} h_\HH \ge 0$

Proof

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Sources

• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next) $II.3.3$