Maximal and minimal extensions

of a symmetric operator $A$

The operators $\overline{A}$ (the closure of $A$, cf. Closed operator) and $A^{\ast }$ (the adjoint of $A$, cf. Adjoint operator), respectively. All closed symmetric extensions of $A$ occur between these. Equality of the maximal and minimal extensions is equivalent to the self-adjointness of $A$ (cf. Self-adjoint operator) and is a necessary and sufficient condition for the uniqueness of a self-adjoint extension.

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