Maximal and minimal extensions

of a symmetric operator $A$

The operators $\bar A$ (the closure of $A$, cf. Closed operator) and $A^*$ (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.

References

[a1]	M. Reed, B. Simon, "Methods of modern mathematical physics" , 1. Functional analysis , Acad. Press (1972) pp. Chapt. 8