Editing Unbounded operator (section)

== Adjoint ==
The adjoint of an unbounded operator can be defined in two equivalent ways. Let <math>T : D(T) \subseteq H_1 \to H_2</math> be an unbounded operator between Hilbert spaces.

First, it can be defined in a way analogous to how one defines the adjoint of a bounded operator. Namely, the adjoint <math>T^* : D\left(T^*\right) \subseteq H_2 \to H_1</math> of {{mvar|T}} is defined as an operator with the property:
<math display=block>\langle Tx \mid y \rangle_2 = \left \langle x \mid T^*y \right \rangle_1, \qquad x \in D(T).</math>
More precisely, <math>T^* y</math> is defined in the following way. If <math>y \in H_2</math> is such that <math>x \mapsto \langle Tx \mid y \rangle</math> is a continuous linear functional on the domain of {{mvar|T}}, then <math>y</math> is declared to be an element of <math>D\left(T^*\right),</math> and after extending the linear functional to the whole space via the [[Hahn–Banach theorem]], it is possible to find some <math>z</math> in <math>H_1</math> such that
<math display=block>\langle Tx \mid y \rangle_2 = \langle x \mid z \rangle_1, \qquad x \in D(T),</math>
since [[Riesz representation theorem]] allows the continuous dual of the Hilbert space <math>H_1</math> to be identified with the set of linear functionals given by the inner product. This vector <math>z</math> is uniquely determined by <math>y</math> if and only if the linear functional <math>x \mapsto \langle Tx \mid y \rangle</math> is densely defined; or equivalently, if {{mvar|T}} is densely defined. Finally, letting <math>T^* y = z</math> completes the construction of <math>T^*,</math> which is necessarily a linear map. The adjoint <math>T^* y</math> exists if and only if {{mvar|T}} is densely defined.

By definition, the domain of <math>T^*</math> consists of elements <math>y</math> in <math>H_2</math> such that <math>x \mapsto \langle Tx \mid y \rangle</math> is continuous on the domain of {{mvar|T}}. Consequently, the domain of <math>T^*</math> could be anything; it could be trivial (that is, contains only zero).<ref name="BSU-3.2">{{ harvnb |Berezansky|Sheftel|Us|1996| loc=Example 3.2 on page 16 }}</ref> It may happen that the domain of <math>T^*</math> is a closed [[hyperplane]] and <math>T^*</math> vanishes everywhere on the domain.<ref name="RS-252">{{ harvnb |Reed|Simon|1980| loc=page 252 }}</ref><ref name="BSU-3.1">{{harvnb|Berezansky|Sheftel|Us|1996|loc=Example 3.1 on page 15 }}</ref> Thus, boundedness of <math>T^*</math> on its domain does not imply boundedness of {{mvar|T}}. On the other hand, if <math>T^*</math> is defined on the whole space then {{mvar|T}} is bounded on its domain and therefore can be extended by continuity to a bounded operator on the whole space.<ref group="nb">Proof: being closed, the everywhere defined <math>T^*</math> is bounded, which implies boundedness of <math>T^{**},</math> the latter being the closure of {{mvar|T}}. See also {{harv |Pedersen|1989| loc=2.3.11 }} for the case of everywhere defined {{mvar|T}}.</ref> If the domain of <math>T^*</math> is dense, then it has its adjoint <math>T^{**}.</math><ref name="Pedersen-5.1.5" /> A closed densely defined operator {{mvar|T}} is bounded if and only if <math>T^*</math> is bounded.<ref group="nb">Proof: <math>T^{**} = T.</math> So if <math>T^*</math> is bounded then its adjoint {{mvar|T}} is bounded.</ref>

The other equivalent definition of the adjoint can be obtained by noticing a general fact. Define a linear operator <math>J</math> as follows:<ref name="Pedersen-5.1.5">{{ harvnb |Pedersen|1989| loc=5.1.5 }}</ref>
<math display=block>\begin{cases} J: H_1 \oplus H_2 \to H_2 \oplus H_1 \\ J(x \oplus y) = -y \oplus x \end{cases}</math>
Since <math>J</math> is an isometric surjection, it is unitary. Hence: <math>J(\Gamma(T))^{\bot}</math> is the graph of some operator <math>S</math> if and only if {{mvar|T}} is densely defined.<ref name="BSU-12">{{harvnb|Berezansky|Sheftel|Us|1996| loc=page 12}}</ref> A simple calculation shows that this "some" <math>S</math> satisfies:
<math display=block>\langle Tx \mid y \rangle_2 = \langle x \mid Sy \rangle_1,</math>
for every {{mvar|x}} in the domain of {{mvar|T}}. Thus <math>S</math> is the adjoint of {{mvar|T}}.

It follows immediately from the above definition that the adjoint <math>T^*</math> is closed.<ref name="Pedersen-5.1.5" /> In particular, a self-adjoint operator (meaning <math>T = T^*</math>) is closed. An operator {{mvar|T}} is closed and densely defined if and only if <math>T^{**} = T.</math><ref group="nb">Proof: If {{mvar|T}} is closed densely defined then <math>T^*</math> exists and is densely defined. Thus <math>T^{**}</math> exists. The graph of {{mvar|T}} is dense in the graph of <math>T^{**};</math> hence <math>T = T^{**}.</math> Conversely, since the existence of <math>T^{**}</math> implies that that of <math>T^*,</math> which in turn implies {{mvar|T}} is densely defined. Since <math>T^{**}</math> is closed, {{mvar|T}} is densely defined and closed.</ref>

Some well-known properties for bounded operators generalize to closed densely defined operators. The kernel of a closed operator is closed. Moreover, the kernel of a closed densely defined operator <math>T : H_1 \to H_2</math> coincides with the orthogonal complement of the range of the adjoint. That is,<ref>{{ harvnb | Brezis | 1983|p=28}}</ref>
<math display=block>\operatorname{ker}(T) = \operatorname{ran}(T^*)^\bot.</math>
[[von Neumann's theorem]] states that <math>T^* T</math> and <math>T T^*</math> are self-adjoint, and that <math>I + T^* T</math> and <math>I + T T^*</math> both have bounded inverses.<ref>{{harvnb | Yoshida | 1980| p=200 }}</ref> If <math>T^*</math> has trivial kernel, {{mvar|T}} has dense range (by the above identity.) Moreover:

:{{mvar|T}} is surjective if and only if there is a <math>K > 0</math> such that <math>\|f\|_2 \leq K \left\|T^* f\right\|_1</math> for all <math>f</math> in <math>D\left(T^*\right).</math><ref group="nb">If <math>T</math> is surjective then <math>T : (\ker T)^{\bot} \to H_2</math> has bounded inverse, denoted by <math>S.</math> The estimate then follows since
<math display="block">\|f\|_2^2 = \left |\langle TSf \mid f \rangle_2 \right | \leq \|S\| \|f\|_2 \left \|T^*f \right \|_1</math>
Conversely, suppose the estimate holds. Since <math>T^*</math> has closed range, it is the case that <math>\operatorname{ran}(T) = \operatorname{ran}\left(T T^*\right).</math> Since <math>\operatorname{ran}(T)</math> is dense, it suffices to show that <math>T T^*</math> has closed range. If <math>T T^* f_j</math> is convergent then <math> f_j</math> is convergent by the estimate since
<math display="block">\|T^*f_j\|_1^2 = | \langle T^*f_j \mid T^*f_j \rangle_1| \leq \|TT^*f_j\|_2 \|f_j\|_2.</math>

Say, <math>f_j \to g.</math> Since <math>T T^*</math> is self-adjoint; thus, closed, (von Neumann's theorem), <math>T T^* f_j \to T T^* g.</math> QED</ref> (This is essentially a variant of the so-called [[closed range theorem]].) In particular, {{mvar|T}} has closed range if and only if <math>T^*</math> has closed range.

In contrast to the bounded case, it is not necessary that <math>(T S)^* = S^* T^*,</math> since, for example, it is even possible that <math>(T S)^*</math> does not exist.{{Citation needed|date=July 2009}}<!-- Need a concrete example.--> This is, however, the case if, for example, {{mvar|T}} is bounded.<ref>{{harvnb | Yoshida|1980| p= 195}}.</ref>

A densely defined, closed operator {{mvar|T}} is called ''[[normal operator|normal]]'' if it satisfies the following equivalent conditions:<ref name="Pedersen-5.1.11">{{ harvnb |Pedersen|1989| loc=5.1.11 }}</ref>

* <math>T^* T = T T^*</math>;
* the domain of {{mvar|T}} is equal to the domain of <math>T^*,</math> and <math>\|T x\| = \left\|T^* x\right\|</math> for every {{mvar|x}} in this domain;
* there exist self-adjoint operators <math>A, B</math> such that <math>T = A + i B,</math><math>T^* = A - i B,</math> and <math>\|T x\|^2 = \|A x\|^2 + \|B x\|^2</math> for every {{mvar|x}} in the domain of {{mvar|T}}.

Every self-adjoint operator is normal.