EPFL Logo
fr|en
Login
Concept

Normal operator

Summary
In mathematics, especially functional analysis, a normal operator on a complex Hilbert space H is a continuous linear operator N : H → H that commutes with its hermitian adjoint N*, that is: NN* = N*N. Normal operators are important because the spectral theorem holds for them. The class of normal operators is well understood. Examples of normal operators are
  • unitary operators: N* = N−1
  • Hermitian operators (i.e., self-adjoint operators): N* = N
  • Skew-Hermitian operators: N* = −N
  • positive operators: N = MM* for some M (so N is self-adjoint).
A normal matrix is the matrix expression of a normal operator on the Hilbert space Cn. Properties Normal operators are characterized by the spectral theorem. A compact normal operator (in particular, a normal operator on a finite-dimensional linear space) is unitarily diagonalizable. Let T be a bounded operator. The following are equivalent.
  • T is normal.
  • T^* is normal.
  • |T x| =
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
Related publications

Loading

Related people

Loading

Related units

Loading

Related concepts

Loading

Related courses

Loading

Related lectures

Loading

Copyright © 2024 EPFL, all rights reserved