Editing Normal operator (section)

==Properties==

Normal operators are characterized by the [[spectral theorem]]. A [[Compact operator on Hilbert space|compact normal operator]] (in particular, a normal operator on a [[dimension (vector space)|finite-dimensional]] [[inner product space]]) is [[unitarily diagonalizable]].{{sfnp|Hoffman|Kunze|1971|page=317}}

Let <math>T</math> be a bounded operator. The following are equivalent.
* <math>T</math> is normal.
* <math>T^{\ast}</math> is normal.
* <math>\|T x\| = \|T^{\ast} x\|</math> for all <math>x</math> (use <math>\|Tx\|^2 = \langle T^{\ast} Tx, x \rangle = \langle T T^*x, x \rangle = \|T^{\ast}x\|^2</math>.
* The self-adjoint and anti–self adjoint parts of <math>T</math> commute. That is, if <math>T</math> is written as <math>T = T_1 + i T_2</math> with <math>T_1 := \frac{T+T^*}{2}</math> and <math>i\,T_2 := \frac{T-T^*}{2},</math> then <math>T_1 T_2 = T_2 T_1.</math><ref group=note>In contrast, for the important class of [[Creation and annihilation operators]] of, e.g., [[quantum field theory]], they don't commute</ref>

If <math>N</math> is a bounded normal operator, then <math>N</math> and <math>N^*</math> have the same kernel and the same range. Consequently, the range of <math>N</math> is dense if and only if <math>N</math> is injective.{{clarify|reason=This follows from a well-known and useful theorem that deserves a link.|date=May 2015}} Put in another way, the kernel of a normal operator is the orthogonal complement of its range. It follows that the kernel of the operator <math>N^k</math> coincides with that of <math>N</math> for any <math>k.</math> Every generalized eigenvalue of a normal operator is thus genuine. <math>\lambda</math> is an eigenvalue of a normal operator <math>N</math> if and only if its complex conjugate <math>\overline{\lambda}</math> is an eigenvalue of <math>N^*.</math> Eigenvectors of a normal operator corresponding to different eigenvalues are orthogonal, and a normal operator stabilizes the orthogonal complement of each of its eigenspaces.<ref name=Naylor>{{cite book|author1=Naylor, Arch W.|author2=Sell George R.|title=Linear Operator Theory in Engineering and Sciences|publisher=Springer|location=New York|year=1982|isbn=978-0-387-95001-3|url=https://books.google.com/books?id=t3SXs4-KrE0C&q=naylor+sell+linear|access-date=2021-06-26|archive-date=2021-06-26|archive-url=https://web.archive.org/web/20210626022510/https://books.google.com/books?id=t3SXs4-KrE0C&q=naylor+sell+linear|url-status=live}}</ref> This implies the usual spectral theorem: every normal operator on a finite-dimensional space is diagonalizable by a unitary operator. There is also an infinite-dimensional version of the spectral theorem expressed in terms of [[projection-valued measure]]s. The residual spectrum of a normal operator is empty.<ref name=Naylor/>

The product of normal operators that commute is again normal; this is nontrivial, but follows directly from [[Fuglede's theorem]], which states (in a form generalized by Putnam):

:If <math>N_1</math> and <math>N_2</math> are normal operators and if <math>A</math> is a bounded linear operator such that <math>N_1 A = A N_2,</math> then <math>N_1^* A = A N_2^*</math>.

The operator norm of a normal operator equals its [[numerical radius]]{{clarify|reason=The link only defines numerical radius for n x n matrices.|date=May 2015}} and [[spectral radius]].

A normal operator coincides with its [[Aluthge transform]].