Normal operator

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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.

A bounded operator T is normal if and only if ||Tx|| = ||T*x|| for all x. ^[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 $C n$ .

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^ 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 $S = T$