Self-adjoint explained

In mathematics, an element of a

is called self-adjoint if it is the same as its adjoint (i.e.

a=a*

).

Definition

Let

l{A}

be a *-algebra. An element

a\inl{A}

is called self-adjoint if

The set of self-adjoint elements is referred to as

l{B}\subseteql{A}

that is closed under the involution *, i.e.

l{B}=l{B}*

, is called

A special case of particular importance is the case where

l{A}

is a complete normed *-algebra, that satisfies the C*-identity (

\left\|a*a\right\|=\left\|a\right\|2\foralla\inl{A}

), 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

l{A}h

,

l{A}H

or

H(l{A})

for the set of self-adjoint elements are also sometimes used, even in the more recent literature.

Examples

a

of a *-algebra, the elements

aa*

and

a*a

are self-adjoint, since * is an

a

of a *-algebra, the real and imaginary parts \operatorname(a) = \frac (a+a^*) and \operatorname(a) = \frac (a-a^*) are self-adjoint, where

i

denotes the

a\inl{A}N

is a normal element of a C*-algebra

l{A}

, 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

l{A}

be a *-algebra. Then:

a\inl{A}

, then

a*a

is self-adjoint, since

(a*a)*=a*(a*)*=a*a

. A similarly calculation yields that

aa*

is also

a=a1a2

be the product of two self-adjoint elements Then

a

is self-adjoint if

a1

and

a2

commutate, since

(a1

*
a
2)

=

*
a
2
*
a
1

=a2a1

always

l{A}

is a C*-algebra, then a normal element

a\inl{A}N

is self-adjoint if and only if its spectrum is real, i.e.

Properties

In *-algebras

Let

l{A}

be a *-algebra. Then:

a\inl{A}

can be uniquely decomposed into real and imaginary parts, i.e. there are uniquely determined elements

a1,a2\inl{A}sa

, so that

a=a1+ia2

holds. Where a_1 = \frac (a + a^*) and

l{A}sa

is a real linear subspace of From the previous property, it follows that

l{A}

is the direct sum of two real linear subspaces, i.e.

a\inl{A}sa

is self-adjoint, then

a

is

l{A}

is called a hermitian *-algebra if every self-adjoint element

a\inl{A}sa

has a real spectrum

In C*-algebras

Let

l{A}

be a C*-algebra and

a\inl{A}sa

. Then:

\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

a+,a-\inl{A}+

, 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

a\inl{A}+

and odd

n\inN

, there exists a uniquely determined

b\inl{A}+

that satisfies

bn=a

, i.e. a unique

n

-th root, as can be shown with the continuous functional

See also

References