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