Editing Unbounded operator (section)

==Extension-related==
{{See also|Extensions of symmetric operators}}

By definition, an operator ''T'' is an ''extension'' of an operator ''S'' if {{math|Γ(''S'') ⊆ Γ(''T'')}}.<ref name="RS-250">{{ harvnb |Reed|Simon|1980| loc=page 250 }}</ref> An equivalent direct definition: for every ''x'' in the domain of ''S'', ''x'' belongs to the domain of ''T'' and {{math|''Sx'' {{=}} ''Tx''}}.<ref name="Pedersen-5.1.1" /><ref name="RS-250" />

Note that an everywhere defined extension exists for every operator, which is a purely algebraic fact explained at {{slink|Discontinuous linear map#General existence theorem}} and based on the [[axiom of choice]]. If the given operator is not bounded then the extension is a [[discontinuous linear map]]. It is of little use since it cannot preserve important properties of the given operator (see below), and usually is highly non-unique.

An operator ''T'' is called ''closable'' if it satisfies the following equivalent conditions:<ref name="Pedersen-5.1.4" /><ref name="RS-250"/><ref name="BSU-6,7">{{ harvnb |Berezansky|Sheftel|Us|1996| loc=pages 6,7 }}</ref>
* ''T'' has a closed extension;
* the closure of the graph of ''T'' is the graph of some operator;
* for every sequence (''x<sub>n</sub>'') of points from the domain of ''T'' such that ''x<sub>n</sub>'' → 0 and also ''Tx<sub>n</sub>'' → ''y'' it holds that {{math|''y'' {{=}} 0}}.

Not all operators are closable.<ref name="BSU-7">{{ harvnb |Berezansky|Sheftel|Us|1996| loc=page 7 }}</ref>

A closable operator ''T'' has the least closed extension <math> \overline T </math> called the ''closure'' of ''T''. The closure of the graph of ''T'' is equal to the graph of <math> \overline T. </math><ref name="Pedersen-5.1.4" /><ref name="RS-250" /> Other, non-minimal closed extensions may exist.<ref name="Pedersen-5.1.16" /><ref name="RS-257-9" />

A densely defined operator ''T'' is closable if and only if ''T''<sup>∗</sup> is densely defined. In this case <math>\overline T = T^{**} </math> and <math> (\overline T)^* = T^*. </math><ref name="Pedersen-5.1.5" /><ref name="RS-253">{{ harvnb |Reed|Simon|1980| loc=page 253 }}</ref>

If ''S'' is densely defined and ''T'' is an extension of ''S'' then ''S''<sup>∗</sup> is an extension of ''T''<sup>∗</sup>.<ref name="Pedersen-5.1.2">{{ harvnb |Pedersen|1989| loc=5.1.2 }}</ref>

Every symmetric operator is closable.<ref name="Pedersen-5.1.6">{{ harvnb |Pedersen|1989| loc=5.1.6 }}</ref>

A symmetric operator is called ''maximal symmetric'' if it has no symmetric extensions, except for itself.<ref name="Pedersen-5.1.3" /> Every self-adjoint operator is maximal symmetric.<ref name="Pedersen-5.1.3" /> The converse is wrong.<ref name="Pedersen-5.2.6">{{ harvnb |Pedersen|1989| loc=5.2.6 }}</ref>

An operator is called ''essentially self-adjoint'' if its closure is self-adjoint.<ref name="Pedersen-5.1.6" /> An operator is essentially self-adjoint if and only if it has one and only one self-adjoint extension.<ref name="RS-256" />

A symmetric operator may have more than one self-adjoint extension, and even a continuum of them.<ref name="RS-257-9" />

A densely defined, symmetric operator ''T'' is essentially self-adjoint if and only if both operators {{math|''T'' – ''i''}}, {{math|''T'' + ''i''}} have dense range.<ref name="RS-257">{{ harvnb |Reed|Simon|1980| loc=page 257 }}</ref>

Let ''T'' be a densely defined operator. Denoting the relation "''T'' is an extension of ''S''" by ''S'' ⊂ ''T'' (a conventional abbreviation for Γ(''S'') ⊆ Γ(''T'')) one has the following.<ref name="RS-255-6">{{ harvnb |Reed|Simon|1980| loc=pages 255, 256 }}</ref>
* If ''T'' is symmetric then ''T'' ⊂ ''T''<sup>∗∗</sup> ⊂ ''T''<sup>∗</sup>.
* If ''T'' is closed and symmetric then ''T'' = ''T''<sup>∗∗</sup> ⊂ ''T''<sup>∗</sup>.
* If ''T'' is self-adjoint then ''T'' = ''T''<sup>∗∗</sup> = ''T''<sup>∗</sup>.
* If ''T'' is essentially self-adjoint then ''T'' ⊂ ''T''<sup>∗∗</sup> = ''T''<sup>∗</sup>.