In functional analysis , the normal operator generalizes the concept of the normal matrix from linear algebra .
definition
If a Hilbert space denotes the set of all continuous endomorphisms of , then an operator is called normal if it commutes with its adjoint operator, i.e. if
X
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L.
(
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)
{\ displaystyle {\ mathcal {L}} (X)}
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{\ displaystyle X}
A.
∈
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(
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{\ displaystyle A \ in {\ mathcal {L}} (X)}
A.
∗
{\ displaystyle A ^ {\ ast}}
A.
A.
∗
=
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∗
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{\ displaystyle AA ^ {\ ast} = A ^ {\ ast} A}
applies.
Examples
properties
Be a normal operator. Then:
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{\ displaystyle A \ in {\ mathcal {L}} (X)}
‖
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x
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=
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∗
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{\ displaystyle \ | Ax \ | = \ | A ^ {\ ast} x \ |}
for all
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∈
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{\ displaystyle x \ in X}
‖
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x
‖
2
≤
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2
x
‖
‖
x
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{\ displaystyle \ | Ax \ | ^ {2} \ leq \ | A ^ {2} x \ | \ | x \ |}
for all
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∈
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{\ displaystyle x \ in X}
The operator norm of is equal to the spectral radius : where denotes the spectrum of .
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{\ displaystyle A}
‖
A.
‖
=
sup
{
|
λ
|
:
λ
∈
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(
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)
}
.
{\ displaystyle \ | A \ | = \ sup \ {| \ lambda | \ colon \ lambda \ in \ sigma (A) \}.}
σ
(
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{\ displaystyle \ sigma (A)}
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{\ displaystyle A}
Of generated C * -algebra and of produced Von Neumann algebra are commutative. This fact enables a functional calculus .
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{\ displaystyle A}
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{\ displaystyle A}
The diagonalizability of normal matrices in linear algebra is generalized to normal operators in the form of the spectral theorem .
A classification of normal operators exists with respect to unitary equivalence modulo compact operators by going over to Calkin's algebra , which is in the finite-dimensional case . This is explained in the article on Calkin's algebra.
{
0
}
{\ displaystyle \ {0 \}}
A bounded operator in a complex Hilbert space can be broken down into with the “real part” and the “imaginary part” . The operators are self-adjoint . is normal if and only if .
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{\ displaystyle A}
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=
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1
+
i
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{\ displaystyle A = W_ {1} + i \, W_ {2}}
W.
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=
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+
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{\ displaystyle W_ {1} = {\ tfrac {1} {2}} (A + A ^ {\ ast})}
W.
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=
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i
(
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-
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)
.
{\ displaystyle W_ {2} = {\ tfrac {1} {2i}} (AA ^ {\ ast}).}
W.
i
{\ displaystyle W_ {i}}
A.
{\ displaystyle A}
W.
1
W.
2
=
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2
W.
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{\ displaystyle W_ {1} W_ {2} = W_ {2} W_ {1}}
Related terms
An operator is called
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{\ displaystyle A \ in {\ mathcal {L}} (X)}
quasinormal , if swapped with , that is .
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{\ displaystyle A \, \!}
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∗
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{\ displaystyle A ^ {\ ast} A}
A.
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∗
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=
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∗
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{\ displaystyle AA ^ {\ ast} A = A ^ {\ ast} AA}
subnormal if there is a Hilbert space such that is subspace of and a normal operator such that and
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{\ displaystyle Y}
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{\ displaystyle X}
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{\ displaystyle Y}
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∈
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{\ displaystyle B \ in {\ mathcal {L}} (Y)}
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⊂
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{\ displaystyle B (X) \ subset X}
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=
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|
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{\ displaystyle A = B | _ {X} \, \!}
hyponormal if for all .
‖
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∗
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≤
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{\ displaystyle \ | A ^ {\ ast} x \ | \ leq \ | Ax \ |}
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∈
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{\ displaystyle x \ in X}
paranormal , if for everyone .
‖
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≤
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2
x
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‖
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{\ displaystyle \ | Ax \ | ^ {2} \ leq \ | A ^ {2} x \ | \ | x \ |}
x
∈
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{\ displaystyle x \ in X}
normaloid , if operator norm = spectral radius, d. h .: .
‖
A.
‖
=
sup
{
|
λ
|
;
λ
∈
σ
(
A.
)
}
{\ displaystyle \ | A \ | = \ sup \ {| \ lambda |; \ lambda \ in \ sigma (A) \}}
The following implications apply:
normal quasinormal subnormal hyponormal paranormal normaloid.
⇒
{\ displaystyle \ Rightarrow}
⇒
{\ displaystyle \ Rightarrow}
⇒
{\ displaystyle \ Rightarrow}
⇒
{\ displaystyle \ Rightarrow}
⇒
{\ displaystyle \ Rightarrow}
Unlimited operators
An unbounded operator with a domain is called normal if
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:
D.
(
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⊆
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→
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{\ displaystyle A: D (A) \ subseteq X \ to X}
D.
(
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{\ displaystyle D (A)}
‖
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x
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=
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∗
x
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,
∀
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∈
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(
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=
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{\ displaystyle \ | Ax \ | = \ | A ^ {\ ast} x \ |, \ qquad \ forall x \ in D (A) = D (A ^ {\ ast})}
applies. The above-mentioned equivalent characterization of normality shows that it is a generalization of the normality of bounded operators. All self-adjoint operators are normal, because for them applies .
A.
∗
=
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{\ displaystyle A ^ {\ ast} = A}
literature
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