Editing Normal operator

{{Short description|(on a complex Hilbert space) continuous linear operator}}
{{Refimprove|date=June 2011}}
In [[mathematics]], especially [[functional analysis]], a '''normal operator''' on a [[complex number|complex]] [[Hilbert space]] <math>H</math> is a [[continuous function (topology)|continuous]] [[linear operator]] <math>N\colon H\rightarrow H</math> that [[commutator|commutes]] with its [[Hermitian adjoint]] <math>N^{\ast}</math>, that is: <math>N^{\ast}N = NN^{\ast}</math>.<ref>{{citation
 | last1 = Hoffman | first1 = Kenneth
 | last2 = Kunze | first2 = Ray | author2-link = Ray Kunze
 | edition = 2nd
 | location = Englewood Cliffs, N.J.
 | mr = 0276251
 | page = 312
 | publisher = Prentice-Hall, Inc.
 | title = Linear algebra
 | year = 1971}}</ref>

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 operator]]s: <math>U^{\ast} = U^{-1}</math>
* [[Hermitian operator]]s (i.e., self-adjoint operators): <math>N^{\ast} = N</math>
* [[skew-Hermitian]] operators: <math>N^{\ast} = -N</math>
* [[positive operator]]s: <math>N = M^{\ast}M</math> for some <math>M</math> (so ''N'' is self-adjoint).

A [[normal matrix]] is the matrix expression of a normal operator on the Hilbert space <math>\mathbb{C}^{n}</math>.

==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]].

==Properties in finite-dimensional case==

If a normal operator ''T'' on a ''finite-dimensional'' real{{clarify|reason=Normal operators were not defined for real Hilbert spaces although the definition is similar and perhaps should be given.|date=May 2015}} or complex Hilbert space (inner product space) ''H'' stabilizes a subspace ''V'', then it also stabilizes its orthogonal complement ''V''<sup>⊥</sup>. (This statement is trivial in the case where ''T'' is self-adjoint.)

''Proof.'' Let ''P<sub>V</sub>'' be the orthogonal projection onto ''V''. Then the orthogonal projection onto ''V''<sup>⊥</sup> is '''1'''<sub>''H''</sub>−''P<sub>V</sub>''. The fact that ''T'' stabilizes ''V'' can be expressed as ('''1'''<sub>''H''</sub>−''P<sub>V</sub>'')''TP<sub>V</sub>'' = 0, or ''TP<sub>V</sub>'' = ''P<sub>V</sub>TP<sub>V</sub>''. The goal is to show that ''P<sub>V</sub>T''('''1'''<sub>''H''</sub>−''P<sub>V</sub>'') = 0.

Let ''X'' = ''P<sub>V</sub>T''('''1'''<sub>''H''</sub>−''P<sub>V</sub>''). Since (''A'', ''B'') ↦ tr(''AB*'') is an [[inner product]] on the space of endomorphisms of ''H'', it is enough to show that tr(''XX*'') = 0. First it is noted that
:<math>\begin{align}
XX^* &= P_V T(\boldsymbol{1}_H - P_V)^2 T^* P_V \\
&= P_V T(\boldsymbol{1}_H - P_V) T^* P_V \\
&= P_V T T^* P_V - P_V T P_V T^* P_V.
\end {align}</math>

Now using properties of the [[Trace (linear algebra)|trace]] and of orthogonal projections we have:

:<math>\begin{align}
\operatorname{tr}(XX^*) &= \operatorname{tr} \left ( P_VTT^*P_V - P_VTP_VT^*P_V \right ) \\
&= \operatorname{tr}(P_VTT^*P_V) - \operatorname{tr}(P_VTP_VT^*P_V) \\
&= \operatorname{tr}(P_V^2TT^*) - \operatorname{tr}(P_V^2TP_VT^*) \\
&= \operatorname{tr}(P_VTT^*) - \operatorname{tr}(P_VTP_VT^*) \\
&= \operatorname{tr}(P_VTT^*) - \operatorname{tr}(TP_VT^*) && \text{using the hypothesis that } T \text{ stabilizes } V\\
&= \operatorname{tr}(P_VTT^*) - \operatorname{tr}(P_VT^*T) \\
&= \operatorname{tr}(P_V(TT^*-T^*T)) \\
&= 0.
\end{align}</math>

The same argument goes through for compact normal operators in infinite dimensional Hilbert spaces, where one make use of the [[Hilbert-Schmidt inner product]], defined by tr(''AB*'') suitably interpreted.<ref>{{cite journal|author=Andô, Tsuyoshi|year=1963|title=Note on invariant subspaces of a compact normal operator|journal=[[Archiv der Mathematik]]|volume=14|pages=337–340|doi=10.1007/BF01234964|s2cid=124945750}}</ref> However, for bounded normal operators, the orthogonal complement to a stable subspace may not be stable.<ref name=Garrett>{{cite web|author=Garrett, Paul|year=2005|title=Operators on Hilbert spaces|url=http://www.math.umn.edu/~garrett/m/fun/Notes/04a_ops_hsp.pdf|access-date=2011-07-01|archive-date=2011-09-18|archive-url=https://web.archive.org/web/20110918213400/http://www.math.umn.edu/~garrett/m/fun/Notes/04a_ops_hsp.pdf|url-status=live}}</ref> It follows that the Hilbert space cannot in general be spanned by eigenvectors of a normal operator. Consider, for example, the [[bilateral shift]] (or two-sided shift) acting on <math>\ell^2(\mathbb{Z})</math>, which is normal, but has no eigenvalues.

The invariant subspaces of a shift acting on Hardy space are characterized by [[Beurling's theorem]].

==Normal elements of algebras==

The notion of normal operators generalizes to an involutive algebra:

An element <math>x</math> of an involutive algebra is said to be normal if <math>x^{\ast}x = xx^{\ast}</math>.

Self-adjoint and unitary elements are normal.

The most important case is when such an algebra is a [[C*-algebra]].

==Unbounded normal operators==

The definition of normal operators naturally generalizes to some class of unbounded operators. Explicitly, a closed operator ''N'' is said to be normal if 
:<math>N^*N = NN^*.</math>
Here, the existence of the adjoint ''N*'' requires that the domain of ''N'' be dense, and the equality includes the assertion that the domain of ''N*N'' equals that of ''NN*'', which is not necessarily the case in general.

Equivalently normal operators are precisely those for which<ref>Weidmann, Lineare Operatoren in Hilberträumen, Chapter 4, Section 3</ref>
:<math>\|Nx\|=\|N^*x\|\qquad</math> 
with
:<math>\mathcal{D}(N)=\mathcal{D}(N^*).</math>
The spectral theorem still holds for unbounded (normal) operators. The proofs work by reduction to bounded (normal) operators.<ref name=Frei>Alexander Frei, Spectral Measures, Mathematics Stack Exchange, [https://math.stackexchange.com/q/1332154 Existence] {{Webarchive|url=https://web.archive.org/web/20210626022512/https://math.stackexchange.com/questions/1332154/spectral-measures-existence |date=2021-06-26 }}, [https://math.stackexchange.com/q/1112508 Uniqueness] {{Webarchive|url=https://web.archive.org/web/20210626022607/https://math.stackexchange.com/questions/1112508/spectral-measures-uniqueness |date=2021-06-26 }}</ref><ref name="Conway">[[John B. Conway]], A Course in Functional Analysis, Second Edition, Chapter X, Section §4</ref>

==Generalization==

The success of the theory of normal operators led to several attempts for generalization by weakening the commutativity requirement. Classes of operators that include normal operators are (in order of inclusion)
* [[Hyponormal operator]]s
* [[Normaloid]]s
* [[Paranormal operator]]s
* [[Quasinormal operator]]s
* [[Subnormal operator]]s

==See also==

* {{annotated link|Continuous linear operator}}
* {{annotated link|Contraction (operator theory)}}

==Notes==

{{reflist|group=note}}

==References==

{{reflist}}

{{Spectral theory}}
{{Hilbert space}}
{{Functional analysis}}

[[Category:Linear operators]]
[[Category:Operator theory]]