NationMaster - Encyclopedia: Normal operator

In functional analysis, a normal operator on a Hilbert space $H$ is a continuous linear operator $N:Hto H$ that commutes with its hermitian adjoint $N *$ : Functional analysis is that branch of mathematics and specifically of analysis which is concerned with the study of spaces of functions. ... In mathematics, a Hilbert space is an inner product space that is complete with respect to the norm defined by the inner product. ... In topology and related areas of mathematics a continuous function is a morphism between topological spaces; that is, a mapping which preserves the topological structure. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... For an electrical switch that periodically reverses the current see commutator (electric) In mathematics, the commutator gives an indication of how poorly a certain binary operation fails to be commutative. ... In mathematics, the Hermitian adjoint of an linear operator is a matching operator (very similar to the inverse operator in concept) defined over a linear space with inner product. ...

$N,N^*=N^*N.$

The main importance of this concept is that the spectral theorem applies to normal operators. In mathematics, particularly linear algebra and functional analysis, the spectral theorem is a collection of results about linear operators or about matrices. ...

Examples of normal operators:

unitary operators ( $N * = N - 1$ )
Hermitian operators ( $N * = N$ )
normal matrices can be seen as normal operators if one takes the Hilbert space to be $C n$ .

Category: Functional analysis In functional analysis, a unitary operator is a bounded linear operator U on a Hilbert space satisfying U*U=UU*=I where I is the identity operator. ... On a finite-dimensional inner product space, a self-adjoint operator is one that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose. ... A complex square matrix A is a normal matrix iff where A* is the conjugate transpose of A (if A is a real matrix, this is the same as the transpose of A). ...

Results from FactBites:

Normal (295 words)
	In behavior etc.: normal means not deviating very much from the average; 'not normal' is often used in a negative sense (improper, sick, etc.).
	In algebra (in particular, group theory): a normal subgroup is a subgroup that is invariant under conjugation.
	In functional analysis: a normal operator is a linear operator on a Hilbert space that commutes with its adjoint.

SPECTRAL DERIVATIVES (2138 words)
	The structure of time t in QM is one of a universal Newtonian variable that is algebraically "central", i.e., considered as a legitimate operator, which it is not, it commutes with all operators, and can therefore be said conceptually to be in the center of the algebra of observables.
	Accommodating the structure of the Heisenberg picture where the operators are time dependent and not the Hilbert space vectors means that one should also be able to extend the notion of spectral derivative to express the spectral derivative of an operator with respect to another operator.
	The operator A is then always diagonalizable by a similarity transformation and has, in general, complex eigenvalues and orthogonal eigenvectors which can be normalized in the usual way [Kato 1966], p.

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