# Self-adjoint element

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Self-adjoint_element
> Markdown URL: https://mediated.wiki/source/Self-adjoint_element.md
> Source: https://en.wikipedia.org/wiki/Self-adjoint_element
> Source revision: 1355094475
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

Element of *-algebra where x* equals x

In [mathematics](/source/Mathematics), an [element](/source/Element_(mathematics)) of a [*-algebra](/source/*-algebra) is called **self-adjoint** if it is the same as its [adjoint](/source/Adjoint_operator) (i.e. a = a ∗ {\displaystyle a=a^{*}} ).

## Definition

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^{*}} .[1]

The [set](/source/Set_(mathematics)) of self-adjoint elements is referred to as A s a {\displaystyle {\mathcal {A}}_{sa}} .

A [subset](/source/Subset) B ⊆ A {\displaystyle {\mathcal {B}}\subseteq {\mathcal {A}}} that is [closed](/source/Closed_set) under the [involution](/source/Involution_(mathematics)) *, i.e. B = B ∗ {\displaystyle {\mathcal {B}}={\mathcal {B}}^{*}} , is called self-adjoint.[2]

A special case of particular importance is the case where A {\displaystyle {\mathcal {A}}} is a [complete normed *-algebra](/source/Banach_algebra#Banach_*-algebras), 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 [C*-algebra](/source/C*-algebra).

Especially in the older literature on *-algebras and C*-algebras, such elements are often called hermitian.[1] 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.

## Examples

- Each [positive element](/source/Positive_element) of a C*-algebra is self-adjoint.[3]

- 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 [involutive antiautomorphism](/source/Antihomomorphism#Involutions).[4]

- For each element a {\displaystyle a} of a *-algebra, the [real and imaginary parts](/source/Real_and_imaginary_parts) 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 [imaginary unit](/source/Imaginary_unit).[1]

- If a ∈ A N {\displaystyle a\in {\mathcal {A}}_{N}} is a [normal element](/source/Normal_element) of a C*-algebra A {\displaystyle {\mathcal {A}}} , then for every [real-valued function](/source/Real-valued_function) f {\displaystyle f} , which is [continuous](/source/Continuous_function) on the [spectrum](/source/Banach_algebra#Spectral_theory) of a {\displaystyle a} , the [continuous functional calculus](/source/Continuous_functional_calculus) defines a self-adjoint element f ( a ) {\displaystyle f(a)} .[5]

## Criteria

Let A {\displaystyle {\mathcal {A}}} be a *-algebra. Then:

- 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.[6]

- Let a = a 1 a 2 {\displaystyle a=a_{1}a_{2}} be the [product](/source/Product_(mathematics)) 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}} [commutate](/source/Commutative_property), 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.[1]

- 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} } .[5]

## Properties

### In *-algebras

Let A {\displaystyle {\mathcal {A}}} be a *-algebra. Then:

- 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^{*})} .[1]

- The set of self-adjoint elements A s a {\displaystyle {\mathcal {A}}_{sa}} is a [real](/source/Real_number) [linear subspace](/source/Linear_subspace) of A {\displaystyle {\mathcal {A}}} . From the previous property, it follows that A {\displaystyle {\mathcal {A}}} is the [direct sum](/source/Direct_sum) 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}} .[7]

- If a ∈ A s a {\displaystyle a\in {\mathcal {A}}_{sa}} is self-adjoint, then a {\displaystyle a} is normal.[1]

- 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} } .[8]

### In C*-algebras

Let A {\displaystyle {\mathcal {A}}} be a C*-algebra and a ∈ A s a {\displaystyle a\in {\mathcal {A}}_{sa}} . Then:

- 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 [spectral radius](/source/Banach_algebra#Spectral_theory), because a {\displaystyle a} is normal.[9]

- 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.[10] The elements a + {\displaystyle a_{+}} and a − {\displaystyle a_{-}} are also referred to as the [positive and negative parts](/source/Positive_and_negative_parts). 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}}} .[11]

- 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} [-th root](/source/Nth_root), as can be shown with the continuous functional calculus.[12]

## See also

- [Self-adjoint matrix](/source/Hermitian_matrix)

- [Self-adjoint operator](/source/Self-adjoint_operator)

## Notes

1. ^ [***a***](#cite_ref-FOOTNOTEDixmier19774_1-0) [***b***](#cite_ref-FOOTNOTEDixmier19774_1-1) [***c***](#cite_ref-FOOTNOTEDixmier19774_1-2) [***d***](#cite_ref-FOOTNOTEDixmier19774_1-3) [***e***](#cite_ref-FOOTNOTEDixmier19774_1-4) [***f***](#cite_ref-FOOTNOTEDixmier19774_1-5) [Dixmier 1977](#CITEREFDixmier1977), p. 4.

1. **[^](#cite_ref-FOOTNOTEDixmier19773_2-0)** [Dixmier 1977](#CITEREFDixmier1977), p. 3.

1. **[^](#cite_ref-FOOTNOTEPalmer2001800_3-0)** [Palmer 2001](#CITEREFPalmer2001), p. 800.

1. **[^](#cite_ref-FOOTNOTEDixmier19773–4_4-0)** [Dixmier 1977](#CITEREFDixmier1977), pp. 3–4.

1. ^ [***a***](#cite_ref-FOOTNOTEKadisonRingrose1983271_5-0) [***b***](#cite_ref-FOOTNOTEKadisonRingrose1983271_5-1) [Kadison & Ringrose 1983](#CITEREFKadisonRingrose1983), p. 271.

1. **[^](#cite_ref-FOOTNOTEPalmer2001798–800_6-0)** [Palmer 2001](#CITEREFPalmer2001), pp. 798–800.

1. **[^](#cite_ref-FOOTNOTEPalmer2001798_7-0)** [Palmer 2001](#CITEREFPalmer2001), p. 798.

1. **[^](#cite_ref-FOOTNOTEPalmer20011008_8-0)** [Palmer 2001](#CITEREFPalmer2001), p. 1008.

1. **[^](#cite_ref-FOOTNOTEKadisonRingrose1983238_9-0)** [Kadison & Ringrose 1983](#CITEREFKadisonRingrose1983), p. 238.

1. **[^](#cite_ref-FOOTNOTEKadisonRingrose1983246_10-0)** [Kadison & Ringrose 1983](#CITEREFKadisonRingrose1983), p. 246.

1. **[^](#cite_ref-FOOTNOTEDixmier197715_11-0)** [Dixmier 1977](#CITEREFDixmier1977), p. 15.

1. **[^](#cite_ref-FOOTNOTEBlackadar200663_12-0)** [Blackadar 2006](#CITEREFBlackadar2006), p. 63.

## References

- Blackadar, Bruce (2006). *Operator Algebras. Theory of C*-Algebras and von Neumann Algebras*. Berlin/Heidelberg: Springer. p. 63. [ISBN](/source/ISBN_(identifier)) [3-540-28486-9](https://en.wikipedia.org/wiki/Special:BookSources/3-540-28486-9).

- [Dixmier, Jacques](/source/Jacques_Dixmier) (1977). *C*-algebras*. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. [ISBN](/source/ISBN_(identifier)) [0-7204-0762-1](https://en.wikipedia.org/wiki/Special:BookSources/0-7204-0762-1). English translation of *Les C*-algèbres et leurs représentations* (in French). Gauthier-Villars. 1969.

- Kadison, Richard V.; Ringrose, John R. (1983). *Fundamentals of the Theory of Operator Algebras. Volume 1 Elementary Theory*. New York/London: Academic Press. [ISBN](/source/ISBN_(identifier)) [0-12-393301-3](https://en.wikipedia.org/wiki/Special:BookSources/0-12-393301-3).

- Palmer, Theodore W. (2001). *Banach algebras and the general theory of*-algebras: Volume 2,*-algebras*. Cambridge university press. [ISBN](/source/ISBN_(identifier)) [0-521-36638-0](https://en.wikipedia.org/wiki/Special:BookSources/0-521-36638-0).

v t e Spectral theory and *-algebras Basic concepts Involution/*-algebra Banach algebra B*-algebra C*-algebra Noncommutative topology Projection-valued measure Spectrum Spectrum of a C*-algebra Spectral radius Operator space Main results Gelfand–Mazur theorem Gelfand–Naimark theorem Gelfand representation Polar decomposition Singular value decomposition Spectral theorem Spectral theory of normal C*-algebras Special Elements/Operators Isospectral Normal operator Hermitian/Self-adjoint operator Unitary operator Unit Spectrum Krein–Rutman theorem Normal eigenvalue Spectrum of a C*-algebra Spectral radius Spectral asymmetry Spectral gap Decomposition Decomposition of a spectrum Continuous Point Residual Approximate point Compression Direct integral Discrete Spectral abscissa Spectral Theorem Borel functional calculus Min-max theorem Positive operator-valued measure Projection-valued measure Riesz projector Rigged Hilbert space Spectral theorem Spectral theory of compact operators Spectral theory of normal C*-algebras Special algebras Amenable Banach algebra With an Approximate identity Banach function algebra Disk algebra Nuclear C*-algebra Uniform algebra Von Neumann algebra Tomita–Takesaki theory Finite-Dimensional Alon–Boppana bound Bauer–Fike theorem Numerical range Schur–Horn theorem Generalizations Dirac spectrum Essential spectrum Pseudospectrum Structure space (Shilov boundary) Miscellaneous Abstract index group Banach algebra cohomology Cohen–Hewitt factorization theorem Extensions of symmetric operators Fredholm theory Limiting absorption principle Schröder–Bernstein theorems for operator algebras Sherman–Takeda theorem Unbounded operator Examples Wiener algebra Applications Almost Mathieu operator Corona theorem Hearing the shape of a drum (Dirichlet eigenvalue) Heat kernel Kuznetsov trace formula Lax pair Proto-value function Ramanujan graph Rayleigh–Faber–Krahn inequality Spectral geometry Spectral method Spectral theory of ordinary differential equations Sturm–Liouville theory Superstrong approximation Transfer operator Transform theory Weyl law Wiener–Khinchin theorem

---
Adapted from the Wikipedia article [Self-adjoint element](https://en.wikipedia.org/wiki/Self-adjoint_element) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Self-adjoint_element?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.