Normal operator - Citizendia

In mathematics, especially functional analysis, a normal operator on a Hilbert space $H$ (or more generally in a C* algebra) is a continuous linear operator

$N:H\to H$

that commutes with its hermitian adjoint N*:

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

Normal operators are characterized by the spectral theorem. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and For functional analysis as used in psychology see the Functional analysis (psychology article This article assumes some familiarity with Analytic geometry and the concept of a limit. C*-algebras (pronounced "C-star" are an important area of research in Functional analysis, a branch of Mathematics. In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that In Mathematics, the commutator gives an indication of the extent to which a certain Binary operation fails to be Commutative. In Mathematics, specifically in Functional analysis, each Linear operator on a Hilbert space has a corresponding adjoint operator. In Mathematics, particularly Linear algebra and Functional analysis, the spectral theorem is any of a number of results about Linear operators

A bounded operator T is normal if and only if ||Tx|| = ||T*x|| for all x. In Functional analysis (a branch of Mathematics) a bounded linear operator is a Linear transformation L between Normed vector spaces ^[1] If N is a normal operator, then N and N* have the same kernel and range. Consequently, the range of N is dense if and only if N is injective.

Examples of normal operators:

unitary operators ( $N * = N - 1$ )
Hermitian operators ( $N * = N$ )
positive operators ( $N = M M *$ )
orthogonal projection operators ( $N = N * = N 2$ )
normal matrices can be seen as normal operators if one takes the Hilbert space to be $C n$ . In Functional analysis, a branch of Mathematics, a unitary operator is a Bounded linear operator U H → In Mathematics, on a finite-dimensional Inner product space, a self-adjoint operator is one that is its own adjoint, or equivalently one whose matrix In Mathematics, especially Functional analysis, a Hermitian element A of a C*-algebra is a positive element if its spectrum A complex square matrix A is a normal matrix if A*A=AA* where A * is the Conjugate transpose

Notes

^ We have $\|Tx\|^2 = \langle T^*Tx, x \rangle$ , $\|T^*x\|^2 = \langle TT^*x, x \rangle$ and the fact that $\langle Tx, x \rangle = \langle Sx, x \rangle$ for all $x$ implies that $S = T$

In Operator theory, quasinormal operators is a class of bounded operators defined by weakening the requirements of a Normal operator. In Mathematics, especially Operator theory, subnormal operators are bounded operators on a Hilbert space defined by weakening the requirements for Normal operators

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