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Difference between revisions of "Maximal and minimal extensions"

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''of a [[Symmetric operator|symmetric operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062910/m0629101.png" />''
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''of a [[Symmetric operator|symmetric operator]] $A$''
  
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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The operators $\bar A$ (the closure of $A$, cf. [[Closed operator|Closed operator]]) and $A^*$ (the adjoint of $A$, cf. [[Adjoint operator|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|Self-adjoint operator]]) and is a necessary and sufficient condition for the uniqueness of a self-adjoint extension.
  
  

Latest revision as of 08:13, 1 August 2014

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.


Comments

References

[a1] M. Reed, B. Simon, "Methods of modern mathematical physics" , 1. Functional analysis , Acad. Press (1972) pp. Chapt. 8
How to Cite This Entry:
Maximal and minimal extensions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Maximal_and_minimal_extensions&oldid=32628
This article was adapted from an original article by A.I. LoginovV.S. Shul'man (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article