View source for Maximal and minimal extensions

''of a [[Symmetric operator|symmetric operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062910/m0629101.png" />''

The operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062910/m0629102.png" /> (the closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062910/m0629103.png" />, cf. [[Closed operator|Closed operator]]) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062910/m0629104.png" /> (the adjoint of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062910/m0629105.png" />, cf. [[Adjoint operator|Adjoint operator]]), respectively. All closed symmetric extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062910/m0629106.png" /> occur between these. Equality of the maximal and minimal extensions is equivalent to the self-adjointness of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062910/m0629107.png" /> (cf. [[Self-adjoint operator|Self-adjoint operator]]) and is a necessary and sufficient condition for the uniqueness of a self-adjoint extension.



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Reed,   B. Simon,   "Methods of modern mathematical physics" , '''1. Functional analysis''' , Acad. Press  (1972)  pp. Chapt. 8</TD></TR></table>