SelfAdjointOperator - crowlogic/arb4j GitHub Wiki

For a self-adjoint operator $T$ on a Hilbert space $\mathcal{H}$, the orthogonal complement of its range is equal to its kernel. This result arises from the general properties of adjoint operators in Hilbert spaces.

Explanation:

1. Orthogonal Complement: The orthogonal complement of a subset $E \subset \mathcal{H}$ is defined as:
   $E^\perp = {x \in \mathcal{H} : \langle x, y \rangle = 0 \text{ for all } y \in E}$.

2. Key Property: For any bounded linear operator $T$, the kernel of $T^*$ (the adjoint operator) is the orthogonal complement of the range of $T$:

   $\ker(T^*) = (\mathrm{Range}(T))^\perp$.

3. Self-Adjoint Case: Since $T = T^*$ for self-adjoint operators, this simplifies directly to:
   $\ker(T) = (\mathrm{Range}(T))^\perp$.

Conclusion:

For a self-adjoint operator $T$, the kernel and the orthogonal complement of its range are identical, reflecting a fundamental property in operator theory and spectral analysis.