Normal operator

In the functional analysis of the normal operator generalizes the concept of the normal matrix of the linear algebra.

Is a Hilbert space and denote the set of all continuous endomorphisms of so called a normal operator if it commutes with its adjoint operator, so if

Applies.

Examples

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

Properties

Be a normal operator. Then:

• For all
• For all
• The operator norm of is equal to the spectral radius: The spectrum denoted by.
• Of generated C * algebra and generated by Von Neumann algebra are commutative. This fact allows a functional calculus.
• The diagonalizability normal matrices in linear algebra generalized for regular operators in the form of the Spectral Theorem.
• A classification of normal operators is with respect to unitary equivalence modulo compact operators by moving onto the Calkin algebra, which is the finite- dimensional case. This is carried out in the article to Calkin algebra.
• A bounded operator in a complex Hilbert space can be decomposed into the " real part " and the " imaginary " in which the operators are self-adjoint. 's Exactly then normal when.

Related terms

An operator is called

• Quasi- normal if with interchanged, that is.
• Subnormal, if there is a Hilbert space, so that sub-space of, and a standard operator, and so that
• Hypo- normal if for all.
• Paranormal, if for all.
• Normaloid if operator norm = spectral radius, ie: .

There are the following implications:

Normal quasi normal subnormal hypo normal paranormal normaloid.

Unbounded operators

An unbounded operator with domain of definition is called normal if

Applies. The above equivalent characterization of normality shows that it is a generalization of the normality of bounded operators. All self-adjoint operators are normal, as is true for this.