Difference between revisions of "Maximal and minimal extensions"
From Encyclopedia of Mathematics
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| − | ''of a [[Symmetric operator|symmetric operator]] | + | {{TEX|done}} |
| + | ''of a [[Symmetric operator|symmetric operator]] $A$'' | ||
| − | The operators | + | 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
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