Editing Unbounded operator (section)

== Symmetric operators and self-adjoint operators ==
{{main|Self-adjoint operator}}

An operator ''T'' on a Hilbert space is ''symmetric'' if and only if for each ''x'' and ''y'' in the domain of {{mvar|T}} we have <math>\langle Tx \mid y \rangle = \lang x \mid Ty \rang</math>. A densely defined operator {{mvar|T}} is symmetric if and only if it agrees with its adjoint ''T''<sup>∗</sup> restricted to the domain of ''T'', in other words when ''T''<sup>∗</sup> is an extension of {{mvar|T}}.<ref name="Pedersen-5.1.3">{{ harvnb |Pedersen|1989| loc=5.1.3 }}</ref>

In general, if ''T'' is densely defined and symmetric, the domain of the adjoint ''T''<sup>∗</sup> need not equal the domain of ''T''. If ''T'' is symmetric and the domain of ''T'' and the domain of the adjoint coincide, then we say that ''T'' is ''self-adjoint''.<ref>{{ harvnb |Kato|1995| loc=5.3.3 }}</ref>  Note that, when ''T'' is self-adjoint, the existence of the adjoint implies that ''T'' is densely defined and since ''T''<sup>∗</sup> is necessarily closed, ''T'' is closed.

A densely defined operator ''T'' is ''symmetric'', if the subspace {{math|Γ(''T'')}} (defined in a previous section) is orthogonal to its image {{math|''J''(Γ(''T''))}} under ''J'' (where ''J''(''x'',''y''):=(''y'',-''x'')).<ref group="nb">Follows from {{ harv |Pedersen|1989| loc=5.1.5 }} and the definition via adjoint operators.</ref>

Equivalently, an operator ''T'' is ''self-adjoint'' if it is densely defined, closed, symmetric, and satisfies the fourth condition: both operators {{math|''T'' – ''i''}}, {{math|''T'' + ''i''}} are surjective, that is, map the domain of ''T'' onto the whole space ''H''. In other words: for every ''x'' in ''H'' there exist ''y'' and ''z'' in the domain of ''T'' such that {{math|''Ty'' – ''iy'' {{=}} ''x''}} and {{math|''Tz'' + ''iz'' {{=}} ''x''}}.<ref name="Pedersen-5.2.5">{{ harvnb |Pedersen|1989| loc=5.2.5 }}</ref>

An operator ''T'' is ''self-adjoint'', if the two subspaces {{math|Γ(''T'')}}, {{math|''J''(Γ(''T''))}} are orthogonal and their sum is the whole space <math> H \oplus H .</math><ref name="Pedersen-5.1.5" />

This approach does not cover non-densely defined closed operators. Non-densely defined symmetric operators can be defined directly or via graphs, but not via adjoint operators.

A symmetric operator is often studied via its [[Cayley transform]].

An operator ''T'' on a complex Hilbert space is symmetric if and only if the number <math> \langle Tx \mid x \rangle </math> is real for all ''x'' in the domain of ''T''.<ref name="Pedersen-5.1.3" />

A densely defined closed symmetric operator ''T'' is self-adjoint if and only if ''T''<sup>∗</sup> is symmetric.<ref name="RS-256">{{ harvnb |Reed|Simon|1980| loc=page 256 }}</ref> It may happen that it is not.<ref name="Pedersen-5.1.16">{{ harvnb |Pedersen|1989| loc=5.1.16 }}</ref><ref name="RS-257-9">{{ harvnb |Reed|Simon|1980| loc=Example on pages 257-259 }}</ref>

A densely defined operator ''T'' is called ''positive''<ref name="Pedersen-5.1.12">{{ harvnb |Pedersen|1989| loc=5.1.12 }}</ref> (or ''nonnegative''<ref name="BSU-25">{{ harvnb |Berezansky|Sheftel|Us|1996| loc=page 25 }}</ref>) if its quadratic form is nonnegative, that is, <math>\langle Tx \mid x \rangle \ge 0 </math> for all ''x'' in the domain of ''T''. Such operator is necessarily symmetric.

The operator ''T''<sup>∗</sup>''T'' is self-adjoint<ref name="Pedersen-5.1.9">{{ harvnb |Pedersen|1989| loc=5.1.9 }}</ref> and positive<ref name="Pedersen-5.1.12" /> for every densely defined, closed ''T''.

The [[Self-adjoint operator#Spectral theorem|spectral theorem]] applies to self-adjoint operators <ref name="Pedersen-5.3.8">{{ harvnb|Pedersen|1989|loc=5.3.8}}</ref> and moreover, to normal operators,<ref name="BSU-89">{{harvnb |Berezansky|Sheftel|Us|1996|loc=page 89}}</ref><ref name="Pedersen-5.3.19">{{ harvnb |Pedersen|1989| loc=5.3.19 }}</ref> but not to densely defined, closed operators in general, since in this case the spectrum can be empty.<ref name="RS-254-E5">{{ harvnb |Reed|Simon|1980| loc=Example 5 on page 254 }}</ref><ref name="Pedersen-5.2.12">{{ harvnb |Pedersen|1989| loc=5.2.12 }}</ref>

A symmetric operator defined everywhere is closed, therefore bounded,<ref name="Pedersen-5.1.4" /> which is the [[Hellinger–Toeplitz theorem]].<ref name="RS-84">{{ harvnb |Reed|Simon|1980| loc=page 84 }}</ref>