Normal operator

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 ${\displaystyle X}$ ${\displaystyle {\mathcal {L}}(X)}$ ${\displaystyle X}$ ${\displaystyle A\in {\mathcal {L}}(X)}$ ${\displaystyle A^{\ast }}$

${\displaystyle AA^{\ast }=A^{\ast }A}$

applies.

Examples

• Self-adjoint and unitary operators are obviously normal.
• The unilateral shift is an example of a non-normal operator.

properties

Be a normal operator. Then: ${\displaystyle A\in {\mathcal {L}}(X)}$

• ${\displaystyle \|Ax\|=\|A^{\ast }x\|}$ for all ${\displaystyle x\in X}$
• ${\displaystyle \|Ax\|^{2}\leq \|A^{2}x\|\|x\|}$ for all ${\displaystyle x\in X}$
• The operator norm of is equal to the spectral radius : where denotes the spectrum of . ${\displaystyle A}$ ${\displaystyle \|A\|=\sup\{|\lambda |\colon \lambda \in \sigma (A)\}.}$ ${\displaystyle \sigma (A)}$ ${\displaystyle A}$
• Of generated C ^* -algebra and of produced Von Neumann algebra are commutative. This fact enables a functional calculus . ${\displaystyle A}$ ${\displaystyle A}$
• 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. ${\displaystyle \{0\}}$
• 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 . ${\displaystyle A}$ ${\displaystyle A=W_{1}+i\,W_{2}}$ ${\displaystyle W_{1}={\tfrac {1}{2}}(A+A^{\ast })}$ ${\displaystyle W_{2}={\tfrac {1}{2i}}(A-A^{\ast }).}$ ${\displaystyle W_{i}}$ ${\displaystyle A}$ ${\displaystyle W_{1}W_{2}=W_{2}W_{1}}$

Related terms

An operator is called ${\displaystyle A\in {\mathcal {L}}(X)}$

• quasinormal , if swapped with , that is . ${\displaystyle A\,\!}$ ${\displaystyle A^{\ast }A}$ ${\displaystyle AA^{\ast }A=A^{\ast }AA}$
• subnormal if there is a Hilbert space such that is subspace of and a normal operator such that and ${\displaystyle Y}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle B\in {\mathcal {L}}(Y)}$ ${\displaystyle B(X)\subset X}$ ${\displaystyle A=B|_{X}\,\!}$
• hyponormal if for all . ${\displaystyle \|A^{\ast }x\|\leq \|Ax\|}$ ${\displaystyle x\in X}$
• paranormal , if for everyone . ${\displaystyle \|Ax\|^{2}\leq \|A^{2}x\|\|x\|}$ ${\displaystyle x\in X}$
• normaloid , if operator norm = spectral radius, d. h .: . ${\displaystyle \|A\|=\sup\{|\lambda |;\lambda \in \sigma (A)\}}$

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 ${\displaystyle A:D(A)\subseteq X\to X}$ ${\displaystyle D(A)}$

${\displaystyle \|Ax\|=\|A^{\ast }x\|,\qquad \forall x\in D(A)=D(A^{\ast })}$

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 . ${\displaystyle A^{\ast }=A}$

literature

• Harro Heuser : Functional Analysis . BG Teubner, Stuttgart (1986), ISBN 3-519-22206-X .
• Gerald Teschl : Mathematical Methods in Quantum Mechanics , American Mathematical Society, Providence (2009), ISBN 978-0-8218-4660-5 . ( free online version )