Editing Unbounded operator (section)

== Definitions and basic properties ==
Let {{math|''X'', ''Y''}} be [[Banach space]]s.  An '''unbounded operator''' (or simply ''operator'') {{math|''T'' : ''D''(''T'') → ''Y''}} is a [[linear map]] {{mvar|T}} from a linear subspace {{math|''D''(''T'') ⊆ ''X''}}—the domain of {{mvar|T}}—to the space {{math|''Y''}}.<ref name="Pedersen-5.1.1">{{harvnb|Pedersen|1989|loc=5.1.1}}</ref> Contrary to the usual convention, {{mvar|T}} may not be defined on the whole space {{mvar|X}}.

An operator {{mvar|T}} is said to be '''[[closed operator|closed]]''' if its graph {{math|Γ(''T'')}} is a [[closed set]].<ref name="Pedersen-5.1.4">{{ harvnb |Pedersen|1989| loc=5.1.4 }}</ref> (Here, the graph {{math|Γ(''T'')}} is a linear subspace of the [[Direct sum of modules#Direct sum of Hilbert spaces|direct sum]] {{math|''X'' ⊕ ''Y''}}, defined as the set of all pairs {{math|(''x'', ''Tx'')}}, where {{mvar|x}} runs over the domain of {{mvar|T}}&thinsp;.) Explicitly, this means that for every sequence {{math|{''x<sub>n</sub>''} }} of points from the domain of {{mvar|T}} such that {{math|''x<sub>n</sub>'' → ''x''}} and {{math|''Tx<sub>n</sub>'' → ''y''}}, it holds that {{mvar|x}} belongs to the domain of {{mvar|T}} and {{math|''Tx'' {{=}} ''y''}}.<ref name="Pedersen-5.1.4"/> The closedness can also be formulated in terms of the ''graph norm'': an operator {{mvar|T}} is closed if and only if its domain {{math|''D''(''T'')}} is a [[complete space]] with respect to the norm:<ref name="BSU-5">{{ harvnb |Berezansky|Sheftel|Us|1996| loc=page 5 }}</ref>

: <math>\|x\|_T = \sqrt{ \|x\|^2 + \|Tx\|^2 }.</math>

An operator {{mvar|T}} is said to be '''[[densely defined operator|densely defined]]''' if its domain is [[dense set|dense]] in {{mvar|X}}.<ref name="Pedersen-5.1.1" /> This also includes operators defined on the entire space {{mvar|X}}, since the whole space is dense in itself. The denseness of the domain is necessary and sufficient for the existence of the adjoint (if {{math|X}} and {{math|Y}} are Hilbert spaces) and the transpose; see the sections below. 

If {{math|''T'' : ''D''(''T'') → ''Y''}} is closed, densely defined and [[continuous operator|continuous]] on its domain, then its domain is all of {{mvar|X}}.<ref group="nb">Suppose ''f<sub>j</sub>'' is a sequence in the domain of {{mvar|T}} that converges to {{math|''g'' ∈ ''X''}}. Since {{mvar|T}} is uniformly continuous on its domain, ''Tf<sub>j</sub>'' is [[Cauchy sequence|Cauchy]] in {{mvar|Y}}. Thus, {{math|(&thinsp;''f<sub>j</sub>''&thinsp;, ''T&thinsp;f<sub>j</sub>''&thinsp;)}} is Cauchy and so converges to some {{math|(&thinsp;''f''&thinsp;, ''T&thinsp;f''&thinsp;)}} since the graph of {{mvar|T}} is closed. Hence, {{math|&thinsp;''f''&thinsp; {{=}} ''g''}}, and the domain of {{mvar|T}} is closed.</ref>

A densely defined symmetric{{clarify|At this point, symmetric isn’t defined. Maybe we should move the paragraph down.|date=July 2024}} operator {{mvar|T}} on a [[Hilbert space]] {{mvar|H}} is called '''bounded from below''' if {{math|''T'' + ''a''}} is a positive operator for some real number {{mvar|a}}. That is, {{math|⟨''Tx''{{!}}''x''⟩ ≥ −''a'' {{!!}}''x''{{!!}}<sup>2</sup>}} for all {{mvar|x}} in the domain of {{mvar|T}} (or alternatively {{math|⟨''Tx''{{!}}''x''⟩ ≥ ''a'' {{!!}}''x''{{!!}}<sup>2</sup>}} since {{math|''a''}} is arbitrary).<ref name="Pedersen-5.1.12" /> If both {{mvar|T}} and {{math|−''T''}} are bounded from below then {{mvar|T}} is bounded.<ref name="Pedersen-5.1.12" />