Non-Archimedean linear operators and applications

107 Pages·2009·1.052 MB·English

by  Diagana T.

N -A L O
ON RCHIMEDEAN INEAR PERATORS
A
AND PPLICATIONS
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N -A L O
ON RCHIMEDEAN INEAR PERATORS
A
AND PPLICATIONS
TOKA DIAGANA
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Contents
Preface .............................................................. xi
1 Non-ArchimedeanValuedFields ..................................... 1
1.1 Introduction...................................................... 1
1.2 Non-ArchimedeanValuedFields .................................... 2
1.2.1 BasicDefinitions............................................ 2
1.2.2 Thet-VectorSpaceKt ....................................... 5
1.3 ConstructionofQ ................................................ 6
p
1.3.1 Introduction ................................................ 6
1.3.2 TheFieldQ ............................................... 6
p
1.3.3 Convergence ofPowerSeriesoverQ .......................... 8
p
1.4 ConstructionofK((x))............................................. 9
1.5 BibliographicalNotes.............................................. 10
2 Non-ArchimedeanBanachandHilbertSpaces ......................... 11
2.1 Non-ArchimedeanBanachSpaces ................................... 11
2.1.1 BasicDefinitions............................................ 11
2.1.2 ExamplesofNon-ArchimedeanBanachSpaces .................. 12
2.2 FreeBanachSpaces ............................................... 13
2.2.1 Definitions ................................................. 13
2.2.2 Examples .................................................. 13
2.3 Non-ArchimedeanHilbertSpaces.................................... 14
2.3.1 Introduction ................................................ 14
2.3.2 Non-ArchimedeanHilbertSpaces.............................. 14
2.3.3 TheHilbertSpaceEω ×Eω ×...×Eω ........................ 16
1 2 t
2.4 BibliographicalNotes.............................................. 18
3 Non-ArchimedeanBoundedLinearOperators ......................... 19
3.1 Introduction...................................................... 19
3.2 Bounded LinearOperatorsonNon-ArchimedeanBanachSpaces ......... 19
viii TokaDiagana
3.2.1 BasicDefinitions............................................ 20
3.2.2 Examples .................................................. 21
3.2.3 TheBanachAlgebraB(X) .................................... 22
3.2.4 FurtherPropertiesofBounded LinearOperators.................. 22
3.3 Bounded LinearOperatorsonNon-ArchimedeanHilbertSpaces.......... 25
3.3.1 Introduction ................................................ 25
3.3.2 RepresentationofBounded OperatorsByInfiniteMatrices......... 25
3.3.3 ExistenceoftheAdjoint...................................... 26
3.3.4 ExamplesofBounded OperatorswithnoAdjoint................. 28
3.4 PerturbationofBases .............................................. 29
3.4.1 Example ................................................... 31
3.5 Hilbert-SchmidtOperators.......................................... 32
3.5.1 BasicDefinitions............................................ 32
3.5.2 FurtherPropertiesofHilbert-SchmidtOperators.................. 38
3.5.3 CompletelyContinuousOperators ............................. 39
3.5.4 Trace...................................................... 40
3.5.5 Examples .................................................. 40
3.6 OpenProblems ................................................... 42
3.7 BibliographicalNotes.............................................. 42
4 Non-ArchimedeanUnboundedLinearOperators ....................... 43
4.1 Introduction...................................................... 43
4.2 BasicDefinitions.................................................. 43
4.2.1 Example ................................................... 44
4.2.2 ExistenceoftheAdjoint...................................... 45
4.2.3 ExamplesofUnbounded OperatorsWithnoAdjoint .............. 45
4.3 ClosedLinearOperatorsonEω...................................... 46
4.4 DiagonalOperatorsonEω .......................................... 49
4.5 OpenProblems ................................................... 51
4.6 BibliographicalNotes.............................................. 51
5 Non-ArchimedeanBilinearForms.................................... 53
5.1 Introduction...................................................... 53
5.2 BasicDefinitions.................................................. 53
5.2.1 ContinuousLinearFunctionalsonEω .......................... 53
5.2.2 Bounded BilinearFormsonEω×Eω ........................... 55
5.2.3 Unbounded BilinearFormsonEω×Eω......................... 56
5.3 ClosedandClosablenon-Archimedean BilinearForms.................. 58
5.3.1 ClosednessoftheFormSum .................................. 59
5.3.2 Constructionofanon-Archimedean HilbertSpaceUsingQuadratic
Forms ..................................................... 61
Contents ix
5.3.3 FurtherPropertiesoftheClosure............................... 62
5.4 RepresentationofBilinearFormsonEω×Eω byLinearOperators ........ 63
5.5 BibliographicalNotes.............................................. 65
6 FunctionsofLinearOperatorsonEω ................................. 67
6.1 Introduction...................................................... 67
6.2 ProductsandSumsofDiagonalOperators............................. 67
6.3 IntegerPowersofDiagonalOperators ................................ 69
6.4 FunctionsofSelf-AdjointOperators.................................. 72
6.5 FunctionsofSomeSymmetricSquareMatricesOverQ ×Q ........... 73
p p
6.5.1 ThePowersoftheMatrixT ................................... 75
6.5.2 ExponentialoftheMatrixT .................................. 75
6.6 OpenProblems ................................................... 76
6.7 BibliographicalNotes.............................................. 76
7 One-ParameterFamilyofBoundedLinearOperatorsonFreeBanachSpaces 77
7.1 Introduction...................................................... 77
7.2 BasicDefinitions.................................................. 78
7.3 Propertiesofnon-ArchimedeanC -Groups ............................ 79
0
7.4 ExistenceofSolutionstoSomep-adicDifferentialEquations............. 83
7.5 OpenProblems ................................................... 86
7.6 BibliographicalNotes.............................................. 86
References ........................................................... 87
Index................................................................ 91