Self-adjoint

<h2 id="definition">Definition</h2>
Let 
 
 
 
 
 
 A
 
 
 
 
 {\displaystyle {\mathcal {A}}}
 
 be a *-algebra. An element 
 
 
 
 a
 ∈
 
 
 A
 
 
 
 
 {\displaystyle a\in {\mathcal {A}}}
 
 is called self-adjoint if 
 
 
 
 a
 =
 
 a
 
 ∗
 
 
 
 
 {\displaystyle a=a^{*}}
 
.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a>
The <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> of self-adjoint elements is referred to as 
 
 
 
 
 
 
 A
 
 
 
 s
 a
 
 
 
 
 {\displaystyle {\mathcal {A}}_{sa}}
 
.
A <a href="/facts/Subset/SJGERmbA">subset</a> 
 
 
 
 
 
 B
 
 
 ⊆
 
 
 A
 
 
 
 
 {\displaystyle {\mathcal {B}}\subseteq {\mathcal {A}}}
 
 that is <a href="/facts/Closed_set/upYnRTUj">closed</a> under the <a href="/facts/Involution_(mathematics)/kKxYrUVa">involution</a> *, i.e. 
 
 
 
 
 
 B
 
 
 =
 
 
 
 B
 
 
 
 ∗
 
 
 
 
 {\displaystyle {\mathcal {B}}={\mathcal {B}}^{*}}
 
, is called self-adjoint.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>
A special case of particular importance is the case where 
 
 
 
 
 
 A
 
 
 
 
 {\displaystyle {\mathcal {A}}}
 
 is a <a href="/facts/Banach_algebra/3BTHdpg2">complete normed *-algebra</a>, that satisfies the C*-identity (
 
 
 
 
 ‖
 
 
 a
 
 ∗
 
 
 a
 
 ‖
 
 =
 
 
 ‖
 a
 ‖
 
 
 2
 
 
  
 ∀
 a
 ∈
 
 
 A
 
 
 
 
 {\displaystyle \left\|a^{*}a\right\|=\left\|a\right\|^{2}\ \forall a\in {\mathcal {A}}}
 
), which is called a <a href="/facts/C*-algebra/B3tnGHIo">C*-algebra</a>.
Especially in the older literature on *-algebras and C*-algebras, such elements are often called hermitian.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> Because of that the notations 
 
 
 
 
 
 
 A
 
 
 
 h
 
 
 
 
 {\displaystyle {\mathcal {A}}_{h}}
 
, 
 
 
 
 
 
 
 A
 
 
 
 H
 
 
 
 
 {\displaystyle {\mathcal {A}}_{H}}
 
 or 
 
 
 
 H
 (
 
 
 A
 
 
 )
 
 
 {\displaystyle H({\mathcal {A}})}
 
 for the set of self-adjoint elements are also sometimes used, even in the more recent literature.

<h2 id="examples">Examples</h2>
<ul><li>Each <a href="/facts/Positive_element/kvTXwFHj">positive element</a> of a C*-algebra is self-adjoint.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a></li>
<li>For each element 
 
 
 
 a
 
 
 {\displaystyle a}
 
 of a *-algebra, the elements 
 
 
 
 a
 
 a
 
 ∗
 
 
 
 
 {\displaystyle aa^{*}}
 
 and 
 
 
 
 
 a
 
 ∗
 
 
 a
 
 
 {\displaystyle a^{*}a}
 
 are self-adjoint, since * is an <a href="/facts/Antihomomorphism/nIRWxknY">involutive antiautomorphism</a>.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a></li>
<li>For each element 
 
 
 
 a
 
 
 {\displaystyle a}
 
 of a *-algebra, the <a href="/facts/Real_and_imaginary_parts/2w7mqI7e">real and imaginary parts</a> 
 
 
 
 Re
 ⁡
 (
 a
 )
 =
 
 
 1
 2
 
 
 (
 a
 +
 
 a
 
 ∗
 
 
 )
 
 
 {\textstyle \operatorname {Re} (a)={\frac {1}{2}}(a+a^{*})}
 
 and 
 
 
 
 Im
 ⁡
 (
 a
 )
 =
 
 
 1
 
 2
 
 i
 
 
 
 
 (
 a
 −
 
 a
 
 ∗
 
 
 )
 
 
 {\textstyle \operatorname {Im} (a)={\frac {1}{2\mathrm {i} }}(a-a^{*})}
 
 are self-adjoint, where 
 
 
 
 
 i
 
 
 
 {\displaystyle \mathrm {i} }
 
 denotes the <a href="/facts/Imaginary_unit/1SCllQlU">imaginary unit</a>.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a></li>
<li>If 
 
 
 
 a
 ∈
 
 
 
 A
 
 
 
 N
 
 
 
 
 {\displaystyle a\in {\mathcal {A}}_{N}}
 
 is a <a href="/facts/Normal_element/RHIIdbaQ">normal element</a> of a C*-algebra 
 
 
 
 
 
 A
 
 
 
 
 {\displaystyle {\mathcal {A}}}
 
, then for every <a href="/facts/Real-valued_function/jetqlJpK">real-valued function</a> 
 
 
 
 f
 
 
 {\displaystyle f}
 
, which is <a href="/facts/Continuous_function/sKbl02pB">continuous</a> on the <a href="/facts/Banach_algebra/3BTHdpg2">spectrum</a> of 
 
 
 
 a
 
 
 {\displaystyle a}
 
, the <a href="/facts/Continuous_functional_calculus/cJ2Nj9Bo">continuous functional calculus</a> defines a self-adjoint element 
 
 
 
 f
 (
 a
 )
 
 
 {\displaystyle f(a)}
 
.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a></li></ul>
<h2 id="criteria">Criteria</h2>
Let 
 
 
 
 
 
 A
 
 
 
 
 {\displaystyle {\mathcal {A}}}
 
 be a *-algebra. Then:

<ul><li>Let 
 
 
 
 a
 ∈
 
 
 A
 
 
 
 
 {\displaystyle a\in {\mathcal {A}}}
 
, then 
 
 
 
 
 a
 
 ∗
 
 
 a
 
 
 {\displaystyle a^{*}a}
 
 is self-adjoint, since 
 
 
 
 (
 
 a
 
 ∗
 
 
 a
 
 )
 
 ∗
 
 
 =
 
 a
 
 ∗
 
 
 (
 
 a
 
 ∗
 
 
 
 )
 
 ∗
 
 
 =
 
 a
 
 ∗
 
 
 a
 
 
 {\displaystyle (a^{*}a)^{*}=a^{*}(a^{*})^{*}=a^{*}a}
 
. A similarly calculation yields that 
 
 
 
 a
 
 a
 
 ∗
 
 
 
 
 {\displaystyle aa^{*}}
 
 is also self-adjoint.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a></li>
<li>Let 
 
 
 
 a
 =
 
 a
 
 1
 
 
 
 a
 
 2
 
 
 
 
 {\displaystyle a=a_{1}a_{2}}
 
 be the <a href="/facts/Product_(mathematics)/aXY11nGS">product</a> of two self-adjoint elements 
 
 
 
 
 a
 
 1
 
 
 ,
 
 a
 
 2
 
 
 ∈
 
 
 
 A
 
 
 
 s
 a
 
 
 
 
 {\displaystyle a_{1},a_{2}\in {\mathcal {A}}_{sa}}
 
. Then 
 
 
 
 a
 
 
 {\displaystyle a}
 
 is self-adjoint if 
 
 
 
 
 a
 
 1
 
 
 
 
 {\displaystyle a_{1}}
 
 and 
 
 
 
 
 a
 
 2
 
 
 
 
 {\displaystyle a_{2}}
 
 <a href="/facts/Commutative_property/WaYxJLxd">commutate</a>, since 
 
 
 
 (
 
 a
 
 1
 
 
 
 a
 
 2
 
 
 
 )
 
 ∗
 
 
 =
 
 a
 
 2
 
 
 ∗
 
 
 
 a
 
 1
 
 
 ∗
 
 
 =
 
 a
 
 2
 
 
 
 a
 
 1
 
 
 
 
 {\displaystyle (a_{1}a_{2})^{*}=a_{2}^{*}a_{1}^{*}=a_{2}a_{1}}
 
 always holds.<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a></li>
<li>If 
 
 
 
 
 
 A
 
 
 
 
 {\displaystyle {\mathcal {A}}}
 
 is a C*-algebra, then a normal element 
 
 
 
 a
 ∈
 
 
 
 A
 
 
 
 N
 
 
 
 
 {\displaystyle a\in {\mathcal {A}}_{N}}
 
 is self-adjoint if and only if its spectrum is real, i.e. 
 
 
 
 σ
 (
 a
 )
 ⊆
 
 R
 
 
 
 {\displaystyle \sigma (a)\subseteq \mathbb {R} }
 
.<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a></li></ul>
<h2 id="properties">Properties</h2>
<h3>In *-algebras</h3>
Let 
 
 
 
 
 
 A
 
 
 
 
 {\displaystyle {\mathcal {A}}}
 
 be a *-algebra. Then:

<ul><li>Each element 
 
 
 
 a
 ∈
 
 
 A
 
 
 
 
 {\displaystyle a\in {\mathcal {A}}}
 
 can be uniquely decomposed into real and imaginary parts, i.e. there are uniquely determined elements 
 
 
 
 
 a
 
 1
 
 
 ,
 
 a
 
 2
 
 
 ∈
 
 
 
 A
 
 
 
 s
 a
 
 
 
 
 {\displaystyle a_{1},a_{2}\in {\mathcal {A}}_{sa}}
 
, so that 
 
 
 
 a
 =
 
 a
 
 1
 
 
 +
 
 i
 
 
 a
 
 2
 
 
 
 
 {\displaystyle a=a_{1}+\mathrm {i} a_{2}}
 
 holds. Where 
 
 
 
 
 a
 
 1
 
 
 =
 
 
 1
 2
 
 
 (
 a
 +
 
 a
 
 ∗
 
 
 )
 
 
 {\textstyle a_{1}={\frac {1}{2}}(a+a^{*})}
 
 and 
 
 
 
 
 a
 
 2
 
 
 =
 
 
 1
 
 2
 
 i
 
 
 
 
 (
 a
 −
 
 a
 
 ∗
 
 
 )
 
 
 {\textstyle a_{2}={\frac {1}{2\mathrm {i} }}(a-a^{*})}
 
.<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a></li>
<li>The set of self-adjoint elements 
 
 
 
 
 
 
 A
 
 
 
 s
 a
 
 
 
 
 {\displaystyle {\mathcal {A}}_{sa}}
 
 is a <a href="/facts/Real_number/R02gw5Pb">real</a> <a href="/facts/Linear_subspace/2wtKTgHL">linear subspace</a> of 
 
 
 
 
 
 A
 
 
 
 
 {\displaystyle {\mathcal {A}}}
 
. From the previous property, it follows that 
 
 
 
 
 
 A
 
 
 
 
 {\displaystyle {\mathcal {A}}}
 
 is the <a href="/facts/Direct_sum/x3ljVsP6">direct sum</a> of two real linear subspaces, i.e. 
 
 
 
 
 
 A
 
 
 =
 
 
 
 A
 
 
 
 s
 a
 
 
 ⊕
 
 i
 
 
 
 
 A
 
 
 
 s
 a
 
 
 
 
 {\displaystyle {\mathcal {A}}={\mathcal {A}}_{sa}\oplus \mathrm {i} {\mathcal {A}}_{sa}}
 
.<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a></li>
<li>If 
 
 
 
 a
 ∈
 
 
 
 A
 
 
 
 s
 a
 
 
 
 
 {\displaystyle a\in {\mathcal {A}}_{sa}}
 
 is self-adjoint, then 
 
 
 
 a
 
 
 {\displaystyle a}
 
 is normal.<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a></li>
<li>The *-algebra 
 
 
 
 
 
 A
 
 
 
 
 {\displaystyle {\mathcal {A}}}
 
 is called a hermitian *-algebra if every self-adjoint element 
 
 
 
 a
 ∈
 
 
 
 A
 
 
 
 s
 a
 
 
 
 
 {\displaystyle a\in {\mathcal {A}}_{sa}}
 
 has a real spectrum 
 
 
 
 σ
 (
 a
 )
 ⊆
 
 R
 
 
 
 {\displaystyle \sigma (a)\subseteq \mathbb {R} }
 
.<a class="footnote-ref" id="fnref:14" href="#fn:14">14</a></li></ul>
<h3>In C*-algebras</h3>
Let 
 
 
 
 
 
 A
 
 
 
 
 {\displaystyle {\mathcal {A}}}
 
 be a C*-algebra and 
 
 
 
 a
 ∈
 
 
 
 A
 
 
 
 s
 a
 
 
 
 
 {\displaystyle a\in {\mathcal {A}}_{sa}}
 
. Then:

<ul><li>For the spectrum 
 
 
 
 
 ‖
 a
 ‖
 
 ∈
 σ
 (
 a
 )
 
 
 {\displaystyle \left\|a\right\|\in \sigma (a)}
 
 or 
 
 
 
 −
 
 ‖
 a
 ‖
 
 ∈
 σ
 (
 a
 )
 
 
 {\displaystyle -\left\|a\right\|\in \sigma (a)}
 
 holds, since 
 
 
 
 σ
 (
 a
 )
 
 
 {\displaystyle \sigma (a)}
 
 is real and 
 
 
 
 r
 (
 a
 )
 =
 
 ‖
 a
 ‖
 
 
 
 {\displaystyle r(a)=\left\|a\right\|}
 
 holds for the <a href="/facts/Banach_algebra/3BTHdpg2">spectral radius</a>, because 
 
 
 
 a
 
 
 {\displaystyle a}
 
 is normal.<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a></li>
<li>According to the continuous functional calculus, there exist uniquely determined positive elements 
 
 
 
 
 a
 
 +
 
 
 ,
 
 a
 
 −
 
 
 ∈
 
 
 
 A
 
 
 
 +
 
 
 
 
 {\displaystyle a_{+},a_{-}\in {\mathcal {A}}_{+}}
 
, such that 
 
 
 
 a
 =
 
 a
 
 +
 
 
 −
 
 a
 
 −
 
 
 
 
 {\displaystyle a=a_{+}-a_{-}}
 
 with 
 
 
 
 
 a
 
 +
 
 
 
 a
 
 −
 
 
 =
 
 a
 
 −
 
 
 
 a
 
 +
 
 
 =
 0
 
 
 {\displaystyle a_{+}a_{-}=a_{-}a_{+}=0}
 
. For the norm, 
 
 
 
 
 ‖
 a
 ‖
 
 =
 max
 (
 
 ‖
 
 a
 
 +
 
 
 ‖
 
 ,
 
 ‖
 
 a
 
 −
 
 
 ‖
 
 )
 
 
 {\displaystyle \left\|a\right\|=\max(\left\|a_{+}\right\|,\left\|a_{-}\right\|)}
 
 holds.<a class="footnote-ref" id="fnref:16" href="#fn:16">16</a> The elements 
 
 
 
 
 a
 
 +
 
 
 
 
 {\displaystyle a_{+}}
 
 and 
 
 
 
 
 a
 
 −
 
 
 
 
 {\displaystyle a_{-}}
 
 are also referred to as the <a href="/facts/Positive_and_negative_parts/Pzy06lhk">positive and negative parts</a>. In addition, 
 
 
 
 
 |
 
 a
 
 |
 
 =
 
 a
 
 +
 
 
 +
 
 a
 
 −
 
 
 
 
 {\displaystyle |a|=a_{+}+a_{-}}
 
 holds for the absolute value defined for every element 
 
 
 
 
 |
 
 a
 
 |
 
 =
 (
 
 a
 
 ∗
 
 
 a
 
 )
 
 
 1
 2
 
 
 
 
 
 {\textstyle |a|=(a^{*}a)^{\frac {1}{2}}}
 
.<a class="footnote-ref" id="fnref:17" href="#fn:17">17</a></li>
<li>For every 
 
 
 
 a
 ∈
 
 
 
 A
 
 
 
 +
 
 
 
 
 {\displaystyle a\in {\mathcal {A}}_{+}}
 
 and odd 
 
 
 
 n
 ∈
 
 N
 
 
 
 {\displaystyle n\in \mathbb {N} }
 
, there exists a uniquely determined 
 
 
 
 b
 ∈
 
 
 
 A
 
 
 
 +
 
 
 
 
 {\displaystyle b\in {\mathcal {A}}_{+}}
 
 that satisfies 
 
 
 
 
 b
 
 n
 
 
 =
 a
 
 
 {\displaystyle b^{n}=a}
 
, i.e. a unique 
 
 
 
 n
 
 
 {\displaystyle n}
 
<a href="/facts/Nth_root/XrRjtJGe">-th root</a>, as can be shown with the continuous functional calculus.<a class="footnote-ref" id="fnref:18" href="#fn:18">18</a></li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Hermitian_matrix/qArPblYo">Self-adjoint matrix</a></li>
<li><a href="/facts/Self-adjoint_operator/VrH1nEAK">Self-adjoint operator</a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li>Blackadar, Bruce (2006). Operator Algebras. Theory of C*-Algebras and von Neumann Algebras. Berlin/Heidelberg: Springer. p. 63. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-28486-9.</li>
<li>Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-7204-0762-1. English translation of Les C*-algèbres et leurs représentations (in French). Gauthier-Villars. 1969.</li>
<li>Kadison, Richard V.; Ringrose, John R. (1983). Fundamentals of the Theory of Operator Algebras. Volume 1 Elementary Theory. New York/London: Academic Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-12-393301-3.</li>
<li>Palmer, Theodore W. (2001). Banach algebras and the general theory of*-algebras: Volume 2,*-algebras. Cambridge university press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-36638-0.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Dixmier 1977, p. 4. - Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Dixmier 1977, p. 3. - Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Dixmier 1977, p. 4. - Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Palmer 2001, p. 800. - Palmer, Theodore W. (2001). Banach algebras and the general theory of*-algebras: Volume 2,*-algebras. Cambridge university press. ISBN 0-521-36638-0. <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Dixmier 1977, pp. 3–4. - Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Dixmier 1977, p. 4. - Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Kadison & Ringrose 1983, p. 271. - Kadison, Richard V.; Ringrose, John R. (1983). Fundamentals of the Theory of Operator Algebras. Volume 1 Elementary Theory. New York/London: Academic Press. ISBN 0-12-393301-3. <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Palmer 2001, pp. 798–800. - Palmer, Theodore W. (2001). Banach algebras and the general theory of*-algebras: Volume 2,*-algebras. Cambridge university press. ISBN 0-521-36638-0. <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Dixmier 1977, p. 4. - Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Kadison & Ringrose 1983, p. 271. - Kadison, Richard V.; Ringrose, John R. (1983). Fundamentals of the Theory of Operator Algebras. Volume 1 Elementary Theory. New York/London: Academic Press. ISBN 0-12-393301-3. <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Dixmier 1977, p. 4. - Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Palmer 2001, p. 798. - Palmer, Theodore W. (2001). Banach algebras and the general theory of*-algebras: Volume 2,*-algebras. Cambridge university press. ISBN 0-521-36638-0. <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Dixmier 1977, p. 4. - Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">Palmer 2001, p. 1008. - Palmer, Theodore W. (2001). Banach algebras and the general theory of*-algebras: Volume 2,*-algebras. Cambridge university press. ISBN 0-521-36638-0. <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">Kadison & Ringrose 1983, p. 238. - Kadison, Richard V.; Ringrose, John R. (1983). Fundamentals of the Theory of Operator Algebras. Volume 1 Elementary Theory. New York/London: Academic Press. ISBN 0-12-393301-3. <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">Kadison & Ringrose 1983, p. 246. - Kadison, Richard V.; Ringrose, John R. (1983). Fundamentals of the Theory of Operator Algebras. Volume 1 Elementary Theory. New York/London: Academic Press. ISBN 0-12-393301-3. <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
<li id="fn:17">Dixmier 1977, p. 15. - Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. <a href="#fnref:17" class="footnote-back-ref">↩</a></li>
<li id="fn:18">Blackadar 2006, p. 63. - Blackadar, Bruce (2006). Operator Algebras. Theory of C*-Algebras and von Neumann Algebras. Berlin/Heidelberg: Springer. p. 63. ISBN 3-540-28486-9. <a href="#fnref:18" class="footnote-back-ref">↩</a></li>
</ol>

Self-adjoint open-in-new

Self-adjoint