Editing Operator theory (section)

==Single operator theory==
Single operator theory deals with the properties and classification of operators, considered one at a time. For example, the classification of [[normal operator]]s in terms of their [[spectrum of an operator|spectra]] falls into this category.

===Spectrum of operators===
{{Main article|Spectral theorem}}

The '''spectral theorem''' is any of a number of results about [[linear operator]]s or about [[matrix (mathematics)|matrices]].<ref>[[V. S. Sunder|Sunder, V.S.]] ''Functional Analysis: Spectral Theory (1997) Birkhäuser Verlag</ref> In broad terms the spectral [[theorem]] provides conditions  under which an [[Operator (mathematics)|operator]] or a matrix can be [[Diagonalizable matrix|diagonalized]] (that is, represented as a [[diagonal matrix]] in some basis).  This concept of diagonalization is relatively straightforward for operators on [[finite-dimensional]] spaces, but requires some modification for operators on infinite-dimensional spaces.  In general, the spectral theorem identifies a class of [[linear operator]]s that can be modelled by [[multiplication operator]]s, which are as simple as one can hope to find. In more abstract language, the spectral theorem is a statement about [[commutative ring|commutative]] [[C*-algebra]]s. See also [[spectral theory]] for a historical perspective.

Examples of operators to which the spectral theorem applies are [[self-adjoint operator]]s or more generally [[normal operator]]s on [[Hilbert space]]s.

The spectral theorem also provides a [[canonical form|canonical]] decomposition, called the '''spectral decomposition''', '''eigenvalue decomposition''', or '''[[eigendecomposition of a matrix|eigendecomposition]]''', of the underlying vector space on which the operator acts.

====Normal operators====
{{main article|Normal operator}}

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>NN^{\ast} = N^{\ast}N</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. Today, 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., selfadjoint operators): <math>N^{\ast} = N</math>; (also, anti-selfadjoint operators: <math>N^{\ast} = -N</math>)
* [[positive operator]]s: <math>N = M^{\ast}M</math>, where <math>M</math> is any operator
* [[Normal matrix|normal matrices]] can be seen as normal operators if one takes the Hilbert space to be <math>\mathbb{C}^{n}</math>.

The spectral theorem extends to a more general class of matrices. Let <math>A</math> be an operator on a finite-dimensional [[inner product space]]. <math>A</math> is said to be [[normal matrix|normal]]  if <math>A^{\ast}A = AA^{\ast}</math>. One can show that <math>A</math> is normal if and only if it is unitarily diagonalizable: By the [[Schur decomposition]], we have <math>A = UTU^{\ast}</math>, where <math>U</math> is unitary and <math>T</math> [[upper triangular]]. Since <math>A</math> is normal, <math>T^{\ast}T = TT^{\ast}</math>. Therefore, <math>T</math> must be diagonal since normal upper triangular matrices are diagonal. The converse is obvious.

In other words, <math>A</math> is normal if and only if there exists a [[unitary matrix]] <math>U</math> such that
<math display="block">A = U D U^* </math>
where <math>D</math> is a [[diagonal matrix]]. Then, the entries of the diagonal of <math>D</math> are the [[eigenvalue]]s of <math>A</math>. The column vectors of <math>U</math> are the [[eigenvector]]s of <math>A</math> and they are [[orthonormal]]. Unlike the Hermitian case, the entries of <math>D</math> need not be real.

===Polar decomposition===
{{Main article|Polar decomposition}}

The '''polar decomposition''' of any [[bounded linear operator]] ''A'' between complex [[Hilbert space]]s is a canonical factorization as the product of a [[partial isometry]] and a non-negative operator.<ref>{{citation|title=A Course in Operator Theory | series=[[Graduate Studies in Mathematics]]|first=John B. |last=Conway|publisher=American Mathematical Society|year= 2000 | isbn=0821820656}}</ref>

The polar decomposition for matrices generalizes as follows: if ''A'' is a bounded linear operator then there is a unique factorization of ''A'' as a product ''A'' = ''UP'' where ''U'' is a partial isometry, ''P'' is a non-negative self-adjoint operator and the initial space of ''U'' is the closure of the range of ''P''.

The operator ''U'' must be weakened to a partial isometry, rather than unitary, because of the following issues. If ''A'' is the [[shift operator|one-sided shift]] on ''l''{{i sup|2}}('''N'''), then |''A''| = (''A*A'')<sup>1/2</sup> = ''I''. So if ''A'' = ''U'' |''A''|, ''U'' must be ''A'', which is not unitary.

The existence of a polar decomposition is a consequence of [[Douglas' lemma]]:
{{math theorem | name = Lemma | math_statement = If ''A'', ''B'' are bounded operators on a Hilbert space ''H'', and ''A*A'' ≤ ''B*B'', then there exists a contraction ''C'' such that ''A'' = ''CB''. Furthermore, ''C'' is unique if ''Ker''(''B*'') ⊂ ''Ker''(''C'').}}

The operator ''C'' can be defined by {{math|1=''C''(''Bh'') = ''Ah''}}, extended by continuity to the closure of ''Ran''(''B''), and by zero on the orthogonal complement of {{math|Ran(''B'')}}. The operator ''C'' is well-defined since {{math|''A*A'' ≤ ''B*B''}} implies {{math|Ker(''B'') ⊂ Ker(''A'')}}. The lemma then follows.

In particular, if {{math|1=''A*A'' = ''B*B''}}, then ''C'' is a partial isometry, which is unique if {{math|Ker(''B*'') ⊂ Ker(''C'').}}
In general, for any bounded operator ''A'',
<math display="block">A^*A = (A^*A)^{\frac{1}{2}} (A^*A)^{\frac{1}{2}},</math>
where (''A*A'')<sup>1/2</sup> is the unique positive square root of ''A*A'' given by the usual [[functional calculus]]. So by the lemma, we have
<math display="block">A = U (A^*A)^{\frac{1}{2}}</math>
for some partial isometry ''U'', which is unique if Ker(''A'') ⊂ Ker(''U''). (Note {{math|1=Ker(''A'') = Ker(''A*A'') = Ker(''B'') = Ker(''B*'')}}, where {{math|1=''B'' = ''B*'' = (''A*A'')<sup>1/2</sup>}}.) Take ''P'' to be (''A*A'')<sup>1/2</sup> and one obtains the polar decomposition ''A'' = ''UP''. Notice that an analogous argument can be used to show ''A = P'U' '', where ''P' '' is positive and ''U' '' a partial isometry.

When ''H'' is finite dimensional, ''U'' can be extended to a unitary operator; this is not true in general (see example above). Alternatively, the polar decomposition can be shown using the operator version of [[singular value decomposition#Bounded operators on Hilbert spaces|singular value decomposition]].

By property of the [[continuous functional calculus]], |''A''| is in the [[C*-algebra]] generated by ''A''. A similar but weaker statement holds for the partial isometry: the polar part ''U'' is in the [[von Neumann algebra]] generated by ''A''. If ''A'' is invertible, ''U'' will be in the [[C*-algebra]] generated by ''A'' as well.

===Connection with complex analysis===
Many operators that are studied are operators on Hilbert spaces of [[holomorphic function]]s, and the study
of the operator is intimately linked to questions in function theory.
For example, [[Beurling's theorem]] describes the [[invariant subspace]]s of the unilateral shift in terms of inner functions, which are bounded holomorphic functions on the unit disk with unimodular boundary values almost everywhere on the circle. Beurling interpreted the unilateral shift as multiplication by the independent variable on the [[Hardy space]].<ref>{{citation|first=Nikolai|last=Nikolski|authorlink = Nikolai Nikolski|title=A treatise on the shift operator|publisher=Springer-Verlag|year=1986| isbn=0-387-90176-0}}. A sophisticated treatment of the connections between Operator theory and Function theory in the [[Hardy space]].</ref>  The success in studying multiplication operators, and more generally [[Toeplitz operator]]s (which are multiplication,  followed by projection onto the Hardy space) has inspired the study of similar questions on other spaces, such as the [[Bergman space]].