Self-adjoint Operator

In mathematics, an element of a *-algebra is called self-adjoint if it is the same as its adjoint (i.e. $a = a^*$).

Definition

Let $\mathcal$ be a *-algebra. An element $a \in \mathcal$ is called self-adjoint if

The set of self-adjoint elements is referred to as

A subset $\mathcal \subseteq \mathcal$ that is closed under the involution *, i.e. $\mathcal = \mathcal^*$, is called

A special case of particular importance is the case where $\mathcal$ is a complete normed *-algebra, that satisfies the C*-identity ($\left\, a^*a \right\, = \left\, a \right\, ^2 \ \forall a \in \mathcal$), which is called a C*-algebra.

Especially in the older literature on *-algebras and C*-algebras, such elements are often called Because of that the notations $\mathcal_h$, $\mathcal_H$ or $H(\mathcal)$ for the set of self-adjoint elements are also sometimes used, even in the more recent literature.

Examples

* Each positive element of a C*-algebra is
* For each element $a$ of a *-algebra, the elements $aa^*$ and $a^*a$ are self-adjoint, since * is an
* For each element $a$ of a *-algebra, the real and imaginary parts $\operatorname(a) = \frac (a+a^*)$ and $\operatorname(a) = \frac (a-a^*)$ are self-adjoint, where $\mathrm$ denotes the
* If $a \in \mathcal_N$ is a normal element of a C*-algebra $\mathcal$, then for every real-valued function $f$, which is continuous on the spectrum of $a$, the continuous functional calculus defines a self-adjoint element

Criteria

Let $\mathcal$ be a *-algebra. Then:

* Let $a \in \mathcal$, then $a^*a$ is self-adjoint, since $(a^*a)^* = a^*(a^*)^* = a^*a$. A similarly calculation yields that $aa^*$ is also
* Let $a = a_1 a_2$ be the product of two self-adjoint elements Then $a$ is self-adjoint if $a_1$ and $a_2$ commutate, since $(a_1 a_2)^* = a_2^* a_1^* = a_2 a_1$ always
* If $\mathcal$ is a C*-algebra, then a normal element $a \in \mathcal_N$ is self-adjoint if and only if its spectrum is real, i.e.

Properties

In *-algebras

Let $\mathcal$ be a *-algebra. Then:

* Each element $a \in \mathcal$ can be uniquely decomposed into real and imaginary parts, i.e. there are uniquely determined elements $a_1,a_2 \in \mathcal_$, so that $a = a_1 + \mathrm a_2$ holds. Where $a_1 = \frac (a + a^*)$ and
* The set of self-adjoint elements $\mathcal_$ is a real linear subspace of From the previous property, it follows that $\mathcal$ is the direct sum of two real linear subspaces, i.e.
* If $a \in \mathcal_$ is self-adjoint, then $a$ is
* The *-algebra $\mathcal$ is called a hermitian *-algebra if every self-adjoint element $a \in \mathcal_$ has a real spectrum

In C*-algebras

Let $\mathcal$ be a C*-algebra and $a \in \mathcal_$. Then:

* For the spectrum $\left\, a \right\, \in \sigma(a)$ or $-\left\, a \right\, \in \sigma(a)$ holds, since $\sigma(a)$ is real and $r(a) = \left\, a \right\,$ holds for the spectral radius, because $a$ is
* According to the continuous functional calculus, there exist uniquely determined positive elements $a_+,a_- \in \mathcal_+$, such that $a = a_+ - a_-$ with For the norm, $\left\, a \right\, = \max(\left\, a_+\right\, ,\left\, a_-\right\, )$ holds. The elements $a_+$ and $a_-$ are also referred to as the positive and negative parts. In addition, $, a, = a_+ + a_-$ holds for the absolute value defined for every element
* For every $a \in \mathcal_+$ and odd $n \in \mathbb$, there exists a uniquely determined $b \in \mathcal_+$ that satisfies $b^n = a$, i.e. a unique $n$ -th root, as can be shown with the continuous functional

See also

* Self-adjoint matrix
* Self-adjoint operator

Notes

References

*
* English translation of
*
*

{{SpectralTheory

Abstract algebra
C*-algebras