Normal operator

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In functional analysis, a normal operator on a Hilbert space <math>H</math> (or more generally in a C* algebra) is a continuous linear operator <math>N:H\to H</math> that commutes with its hermitian adjoint <math>N^*</math>:

<math> N\,N^*=N^*N. </math>

The main importance of this concept is that the spectral theorem applies to normal operators.

Examples of normal operators:

• unitary operators ( <math>N^*=N^{-1}</math> )
• Hermitian operators ( <math>N^*=N</math> )
• positive operators (<math>N = MM^*</math>)
• orthogonal projection operators (<math>N=N^*=N^2</math>)
• normal matrices can be seen as normal operators if one takes the Hilbert space to be <math>C^n</math>.