Wikipedia:WikiProject Mathematics/PlanetMath Exchange/47-XX Operator theory: Difference between revisions

Revision as of 20:09, 3 July 2005

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__NOTOC__

{{Planetmath instructions|topic=47-XX Operator theory}}

==47-00 General reference works (handbooks, dictionaries, bibliographies, etc.)==

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=38639729 Fredholmoperator indextopologies], id=38639729^{<span style="color:red;">new!</span>} -- WP guess: [[Fredholmoperator indextopologies]] -- Status:

==47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)==

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=4522 adjoint], id=4522 -- WP guess: [[adjoint]] -- Status:

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=4321 Baker-Campbell-Hausdorff formula(e)], id=4321 -- WP guess: [[Baker-Campbell-Hausdorff formula(e)]] -- Status: '''A'''

:: WP article looks reasonable. [[User:Oleg Alexandrov|Oleg Alexandrov]] 15:27, 9 September 2005 (UTC)

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=4526 closed operator], id=4526 -- WP guess: [[closed operator]] -- Status: '''C'''

::[[User:Oleg Alexandrov|Oleg Alexandrov]] 19:58, 9 September 2005 (UTC)

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=4523 densely defined], id=4523 -- WP guess: [[densely defined]] -- Status:

Line 12 ⟶ 20:

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=4524 properties of the adjoint operator], id=4524 -- WP guess: [[properties of the adjoint operator]] -- Status:

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=8998075 triplepolar scalar productdecomposition], id=8998075^{<span style="color:red;">new!</span>} -- '''DuplicateWP guess: [[polar decomposition]] -- entry'''.Status:

:See [[Wikipedia:WikiProject Mathematics/PlanetMath Exchange/46-XX Functional analysis#46M05 Tensor products|46M05 Tensor products]]

==47A07 Forms (bilinear, sesquilinear, multilinear)==

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=1612 bilinear form], id=1612 -- '''Duplicate entry'''.

:See [[Wikipedia:WikiProject Mathematics/PlanetMath Exchange/11-XX Number theory#11E39 Bilinear and Hermitian forms|11E39 Bilinear and Hermitian forms]]

==47A10 Spectrum, resolvent==

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=6631 canonical basis for symmetric bilinear forms], id=6631 -- '''Duplicate entry'''.

:See [[Wikipedia:WikiProject Mathematics/PlanetMath Exchange/11-XX Number theory#11E39 Bilinear and Hermitian forms|11E39 Bilinear and Hermitian forms]]

==47A12 Numerical range, numerical radius==

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=2468 Hermitian form], id=2468 -- '''Duplicate entry'''.

:See [[Wikipedia:WikiProject Mathematics/PlanetMath Exchange/11-XX Number theory#11E39 Bilinear and Hermitian forms|11E39 Bilinear and Hermitian forms]]

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=66309212 matrix representationDunkl-Williams of a bilinear forminequality], id=66309212^{<span style="color:red;">new!</span>} -- '''DuplicateWP guess: [[Dunkl-Williams inequality]] -- entry'''.Status:

:See [[Wikipedia:WikiProject Mathematics/PlanetMath Exchange/11-XX Number theory#11E39 Bilinear and Hermitian forms|11E39 Bilinear and Hermitian forms]]

==47A15 Invariant subspaces==

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=2483 non-degenerate bilinear form], id=2483 -- '''Duplicate entry'''.

:See [[Wikipedia:WikiProject Mathematics/PlanetMath Exchange/11-XX Number theory#11E39 Bilinear and Hermitian forms|11E39 Bilinear and Hermitian forms]]

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=2472 positive definite form], id=2472 -- '''Duplicate entry'''.

:See [[Wikipedia:WikiProject Mathematics/PlanetMath Exchange/11-XX Number theory#11E39 Bilinear and Hermitian forms|11E39 Bilinear and Hermitian forms]]

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=2466 symmetric bilinear form], id=2466 -- '''Duplicate entry'''.

:See [[Wikipedia:WikiProject Mathematics/PlanetMath Exchange/11-XX Number theory#11E39 Bilinear and Hermitian forms|11E39 Bilinear and Hermitian forms]]

==47A35 Ergodic theory==

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=1996 ergodic theorem], id=1996 -- '''Duplicate entry'''.

:See [[Wikipedia:WikiProject Mathematics/PlanetMath Exchange/37-XX Dynamical systems and ergodic theory#37A30 Ergodic theorems, spectral theory, Markov operators|37A30 Ergodic theorems, spectral theory, Markov operators]]

==47A53 (Semi-) Fredholm operators; index theories==

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=3863 Fredholm index], id=3863 -- WP guess: [[Fredholm index]] -- Status:

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=33293863 Fredholm moduleindex], id=33293863 -- WP: [[Fredholm index]] -- Status: '''Duplicate entryA'''.

:See [[Wikipedia:WikiProject Mathematics/PlanetMath Exchange/46-XX Functional analysis#46L87 Noncommutative differential geometry|46L87 Noncommutative differential geometry]]

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=3353 Fredholm operator], id=3353 -- WP guess: [[Fredholm operator]] -- Status: '''A'''

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=5737 semi-Fredholm operator], id=5737 -- WP guess: [[semi-Fredholm operator]] -- Status: '''N'''

:: The PM article is a barebone definition. I would not be willing to create an article for that. [[User:Oleg Alexandrov|Oleg Alexandrov]] 15:26, 9 September 2005 (UTC)

==47A55 Perturbation theory==

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=6562 Kato-Rellich theorem], id=6562 -- WP guess: [[Kato-Rellich theorem]] -- Status:

==47A56 Functions whose values are linear operators (operator and matrix valued functions, etc., including analytic and meromorphic ones==

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=4311 Taylor's formula for matrix functions], id=4311 -- WP guess: [[Taylor's formula for matrix functions]] -- Status:

==47A60 Functional calculus==

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=2013 Beltrami identity], id=2013 -- WP guess: [[Beltrami identity]] -- Status:

Line 69 ⟶ 64:

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=2092 Euler-Lagrange differential equation (elementary)], id=2092 -- WP guess: [[Euler-Lagrange differential equation (elementary)]] -- Status:

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=24687457 Hermitianstationary formpoint], id=24687457 -- '''DuplicateWP entry'''.guess: [[stationary point]] -- Status:

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=19968074 ergodicRusso-Dye theorem], id=19968074^{<span style="color:red;">new!</span>} -- '''DuplicateWP guess: [[Russo-Dye theorem]] -- entry'''.Status:

==47Axx General theory of linear operators==

==47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.)==

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=4527 self-adjoint operator], id=4527 -- WP guess: [[self-adjoint operator]] -- Status:

==47B25 Symmetric and selfadjoint operators (unbounded)==

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=6563 basic criterion for self-adjointness], id=6563 -- WP guess: [[basic criterion for self-adjointness]] -- Status:

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=6564 proof of basic criterion for self-adjointness], id=6564 -- WP guess: [[proof of basic criterion for self-adjointness]] -- Status:

==47B38 Operators on function spaces (general)==

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=4527 self-adjoint operator], id=4527 -- '''Duplicate entry'''.

:See [[Wikipedia:WikiProject Mathematics/PlanetMath Exchange/47-XX Operator theory#47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.)|47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.)]]

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=47387654 Drazinmultiplication inverseoperator on L^2], id=47387654^{<span style="color:red;">new!</span>} -- WP guess: [[Drazinmultiplication operator on inverseL^2]] -- Status:

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=62717798 functionaloperator calculusnorm forof Hermitianmultiplication matricesoperator on L^2], id=62717798^{<span style="color:red;">new!</span>} -- WP guess: [[functionaloperator norm of calculusmultiplication foroperator Hermitianon matricesL^2]] -- Status:

==47Bxx Special classes of linear operators==

==47C05 Operators in algebras==

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=6271 functional calculus for Hermitian matrices], id=6271 -- WP guess: [[functional calculus for Hermitian matrices]] -- Status:

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=66316271 canonicalfunctional basiscalculus for symmetricHermitian bilinear formsmatrices], id=66316271 -- '''DuplicateWP guess: [[functional calculus]] -- entry'''.Status:

==47C10 Operators in $^*$-algebras==

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=47147830 Dinipartial derivativeisometry], id=47147830^{<span style="color:red;">new!</span>} -- WP guess: [[Dinipartial derivativeisometry]] -- Status: '''Copied'''.

==47C15 Operators in $C^*$- or von Neumann algebras==

==47Cxx Individual linear operators as elements of algebraic systems==

==47G30 Pseudodifferential operators==

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=4714 Dini derivative], id=4714 -- WP: [[Dini derivative]] -- Status: '''Copied'''.

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=24834714 non-degenerateDini bilinear formderivative], id=24834714 -- WP: [[Dini derivative]] -- Status: '''Duplicate entryC'''.

:: [[User:Oleg Alexandrov|Oleg Alexandrov]] 3 July 2005 20:09 (UTC)

==47Gxx Integral, integro-differential, and pseudodifferential operators==

==47H10 Fixed-point theorems==

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=4705 any topological space with the fixed point property is connected], id=4705 -- WP guess: [[any topological space with the fixed point property is connected]] -- Status:

Line 98 ⟶ 119:

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=3642 proof of Brouwer fixed point theorem], id=3642 -- WP guess: [[proof of Brouwer fixed point theorem]] -- Status:

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=24668597 symmetricKKM bilinear formlemma], id=24668597^{<span style="color:red;">new!</span>} -- '''DuplicateWP guess: [[KKM lemma]] -- entry'''.Status:

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=9274 the topologist's sine curve has the fixed point property], id=9274^{<span style="color:red;">new!</span>} -- WP guess: [[the topologist's sine curve has the fixed point property]] -- Status:

==47Hxx Nonlinear operators and their properties==

==47J07 Abstract inverse mapping and implicit function theorems==

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=5927 Lipschitz inverse mapping theorem], id=5927 -- '''Duplicate entry'''.

==47Jxx Equations and inequalities involving nonlinear operators==

:See [[Wikipedia:WikiProject Mathematics/PlanetMath Exchange/46-XX Functional analysis#46B07 Local theory of Banach spaces|46B07 Local theory of Banach spaces]]

==47L07 Convex sets and cones of operators==

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=4443 convex hull of S is open if S is open], id=4443 -- '''Duplicate entry'''.

:See [[Wikipedia:WikiProject Mathematics/PlanetMath Exchange/46-XX Functional analysis#46A55 Convex sets in topological linear spaces; Choquet theory|46A55 Convex sets in topological linear spaces; Choquet theory]]

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=5587 proof that the convex hull of S is open if S is open], id=5587 -- '''Duplicate entry'''.

:See [[Wikipedia:WikiProject Mathematics/PlanetMath Exchange/46-XX Functional analysis#46A55 Convex sets in topological linear spaces; Choquet theory|46A55 Convex sets in topological linear spaces; Choquet theory]]

==47L25 Operator spaces (= matricially normed spaces)==

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=3018 operator norm], id=3018 -- WP guess: [[operator norm]] -- Status:

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=44437091 convex hullexamples of Sbounded isand openunbounded if S is openoperators], id=44437091 -- WP: [[examples of bounded and unbounded operators]] -- Status: '''Duplicate entryA'''.

:: [[User:Oleg Alexandrov|Oleg Alexandrov]] ([[User talk:Oleg Alexandrov|talk]]) 04:36, 10 March 2006 (UTC)

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=7732 derivative operator is unbounded in the sup norm], id=7732^{<span style="color:red;">new!</span>} -- WP guess: [[derivative operator is unbounded in the sup norm]] -- Status:

==47Lxx Linear spaces and algebras of operators==

==47S99 Miscellaneous==

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=4738 Drazin inverse], id=4738 -- WP guess: [[Drazin inverse]] -- Status:

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=16124738 bilinearDrazin forminverse], id=16124738 -- WP: [[Drazin inverse]] -- Status: '''Duplicate entryN'''.

:Just a definition of a rather obscure kind of inverse for operators which are not normally invertible. [[User:Oleg Alexandrov|Oleg Alexandrov]] 20:08, 9 September 2005 (UTC)

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=24727577 positivematrix definiteinversion formlemma], id=24727577 -- '''DuplicateWP entry'''.guess: [[matrix inversion lemma]] -- Status:

==47Sxx Other (nonclassical) types of operator theory==