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| | Nice to be here there, I am Adrianne and I totally really like that name. Vermont has always been my home and I true love every day living suitable here. Gardening is what I do invest in. I am a people manager only soon I'll be by myself. You can find my website here: http://prometeu.net<br><br>Check out my webpage; how to hack clash of clans, [http://prometeu.net Full Article], |
| In [[mathematics]], especially [[functional analysis]], a '''normal operator''' on a complex [[Hilbert space]] ''H'' is a [[continuous function (topology)|continuous]] [[linear operator]] ''N'' : ''H'' → ''H'' that [[commutator|commutes]] with its [[hermitian adjoint]] ''N*'', that is: ''NN*'' = ''N*N''. <ref>{{cite book|author=Hoffman, Kenneth & Kunze, Ray|year=1971|title=Linear Algebra|edition=Second|pages=312}}</ref>
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| Normal operators are important because the [[spectral theorem]] holds for them. Today, the class of normal operators is well-understood. Examples of normal operators are
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| * [[unitary operator]]s: ''N*'' = ''N<sup>−1</sup>
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| * [[Hermitian operator]]s (i.e., selfadjoint operators): ''N*'' = ''N''; (also, anti-selfadjoint operators: ''N*'' = −''N'')
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| * [[positive operator]]s: ''N'' = ''MM*''<!-- where M stands for what? -->
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| * [[Normal matrix|normal matrices]] can be seen as normal operators if one takes the Hilbert space to be '''C'''<sup>''n''</sup>.
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| == Properties ==
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| Normal operators are characterized by the [[spectral theorem]]. A [[Compact operator on Hilbert space|compact normal operator]] (in particular, a normal operator on a finite-dimensional linear space) is unitarily diagonalizable.<ref>{{cite book|author=Hoffman, Kenneth & Kunze, Ray|year=1971|title=Linear Algebra|edition=Second|pages=317}}</ref>
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| Let ''T'' be a bounded operator. The following are equivalent.
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| *''T'' is normal.
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| *''T*'' is normal.
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| *||''Tx''|| = ||''T*x''|| for all ''x'' (use <math>\|Tx\|^2 = \langle T^*Tx, x \rangle = \langle TT^*x, x \rangle = \|T^*x\|^2</math>).
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| *The selfadjoint and anti-selfadjoint parts of ''T'' (i.e., <math>T\equiv T_1+i T_2,</math> with <math>T_1:=\frac{T+T^*}{2}</math> rsp. <math> i\,T_2:=\frac{T-T^*}{2}\,),</math> commute.<ref>In contrast, for the important class of [[Creation and annihilation operators]] of, e.g., [[quantum field theory]], they don't commute</ref>
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| 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. Put in another way, the kernel of a normal operator is the orthogonal complement of its range; thus, the kernel of the operator ''N<sup>k</sup>'' coincides with that of ''N'' for any ''k''. Every generalized eigenvalue of a normal operator is thus genuine. λ is an eigenvalue of a normal operator ''N'' if and only if its complex conjugate <math>\overline{\lambda}</math> is an eigenvalue of ''N*''. Eigenvectors of a normal operator corresponding to different eigenvalues are orthogonal, and it stabilizes orthogonal complements to its eigenspaces.<ref name=Naylor>{{cite book |author=Naylor, Arch W.; Sell George R.|title=Linear Operator Theory in Engineering and Sciences|publisher=Springer|location=New York|year=1982 |pages= |isbn=978-0-387-95001-3|url=http://books.google.com/books?id=t3SXs4-KrE0C&dq=naylor+sell+linear}}</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 generalization in terms of [[projection-valued measures]]. Residual spectrum of a normal operator is empty.<ref name=Naylor/>
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| The product of normal operators that commute is again normal; this is nontrivial and follows from [[Fuglede's theorem]], which states (in a form generalized by Putnam):
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| :If <math>N_1</math> and <math>N_2</math> are normal operators and if ''A'' is a bounded linear operator such that <math>N_1 A = A N_2</math>, then <math>N_1^* A = A N_2^*</math>. | |
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| The operator norm of a normal operator equals its [[numerical radius]] and [[spectral radius]].
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| A normal operator coincides with its [[Aluthge transform]].
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| == Properties in finite-dimensional case ==
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| If a normal operator ''T'' on a ''finite-dimensional'' real 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 )
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| ''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 ''X'' := ''P<sub>V</sub>T''('''1'''<sub>''H''</sub>−''P<sub>V</sub>'') = 0. 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. But first we express ''XX*'' in terms of orthogonal projections:
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| :<math>XX^* = P_VT(\boldsymbol{1}_H-P_V)^2T^*P_V= P_VT(\boldsymbol{1}_H-P_V)T^*P_V = P_VTT^*P_V - P_VTP_VT^*P_V</math>,
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| Now using properties of the [[Trace (linear algebra)|trace]] and of orthogonal projections we have:
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| :<math>\begin{align}
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| \operatorname{tr}(XX^*) &= \operatorname{tr} \left ( P_VTT^*P_V - P_VTP_VT^*P_V \right ) \\
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| &= \operatorname{tr}(P_VTT^*P_V) - \operatorname{tr}(P_VTP_VT^*P_V) \\
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| &= \operatorname{tr}(P_V^2TT^*) - \operatorname{tr}(P_V^2TP_VT^*) \\
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| &= \operatorname{tr}(P_VTT^*) - \operatorname{tr}(P_VTP_VT^*) \\
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| &= \operatorname{tr}(P_VTT^*) - \operatorname{tr}(TP_VT^*) \\
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| &= \operatorname{tr}(P_VTT^*) - \operatorname{tr}(P_VT^*T) \\
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| &= \operatorname{tr}(P_V(TT^*-T^*T)) \\
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| &= 0.
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| \end{align}</math>
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| The same argument goes through for compact normal operators in infinite dimensional Hilbert spaces, where one make use of the [[Hilbert-Schmidt]] inner product.<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}}</ref> However, for bounded normal operators 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}}</ref> It follows that such subspaces cannot be spanned by eigenvectors. Consider, for example, the [[bilateral shift]], which has no eigenvalues. The invariant subspaces of the bilateral shift is characterized by [[Beurling's theorem]].
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| == Normal elements ==
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| The notion of normal operators generalizes to an involutive algebra; namely, an element ''x'' of an involutive algebra is said to be normal if ''xx*'' = ''x*x''. The most important case is when such an algebra is a [[C*-algebra]]. A [[positive element]] is an example of a normal element.
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| == Unbounded normal operators ==
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| The definition of normal operators naturally generalizes to some class of unbounded operators. Explicitly, a closed operator ''N'' is said to be normal if
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| :<math>N^*N = NN^*</math>
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| Here, the existence of the adjoint ''N*'' implies that the domain of ''N'' is dense, and the equality implies that the domain of ''N*N'' equals that of ''NN*'', which is not necessarily the case in general.
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| The spectral theorem still holds for unbounded normal operators, but usually requires a different proof.
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| == Generalization ==
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| 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)
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| *[[Quasinormal operator]]s
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| *[[Subnormal operator]]s
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| *[[Hyponormal operator]]s
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| *[[Paranormal operator]]s
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| *[[Normaloid]]s
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| == Notes ==
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| <references />
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| == References ==
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| * Hoffman, Kenneth and Kunze, Ray. ''Linear Algebra''. Second Edition. 1971. Prentice-Hall, Inc.
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| {{Functional Analysis}}
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| [[Category:Operator theory]]
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Nice to be here there, I am Adrianne and I totally really like that name. Vermont has always been my home and I true love every day living suitable here. Gardening is what I do invest in. I am a people manager only soon I'll be by myself. You can find my website here: http://prometeu.net
Check out my webpage; how to hack clash of clans, Full Article,