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An algebraic equation for linear operators

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An algebraic equation for linear operators

Linear Algebraandits Applications412(2006)303325

    www.elsevier.com/locate/laa

    Analgebraicequationforlinearoperators

    Che Tat Ng

    a

    , Weinian Zhang

    b , ?

    a

    DepartmentofPure Mathematics, UniversityofWaterloo, Waterloo,

    ON,Canada N2L 3G1

    b

    DepartmentofMathematics,Sichuan University,Chengdu,Sichuan 610064,PR China

    Received9 September2004;accepted27June2005

    Availableonline26 August2005

    Submittedby R.A.Brualdi

    Abstract

    Let T bealinearoperatoronavectorspace V ,possiblyofin?nitedimension,overageneral

    ?eld K . Wesolvethefunctionalequation p(T) = F where p ? K [ x ] and F ,an algebraic operatoron V ,aregiven.Fornilpotent F wegiveanexplicitlinearsystem which determines

    thesolutionsbytheirsimilarity classes.The methodisbased on acanonicaldecomposition

    theorem.

    ? 2005 ElsevierInc.Allrightsreserved.

    AMSclassi?cation: 47A62;39B12;15A21;37E05

    Keywords: Functionalequation;Fractionaliterate;Babbage’sequation;Linearoperator;Canonicaldecom-

    position;Normalform; Matrixequation

    1.Introduction

    An iteration isacompositionofafunction withitself.Let n 0beanintegerand

    let T

    n

    denotethe n thiterate of T ,i.e., T

    0

    istheidentity map and T

    k

    = T ? T

    k ? 1

    ,

    where ? denotesthecomposition

    offunctions.Equations withiteration asits

    main

Supportedby NSERC(Canada) GrantRGPIN8212,NSFC(China) Grant#10471101,TRAPOYT,and

    China MOE Research Grants.

    ? Correspondingauthor.

    E-mailaddresses: ctng@math.uwaterloo.ca(C.T.Ng),wnzhang@scu.edu.cn(W.Zhang).

    0024-3795/$-seefront matter ( 2005 ElsevierInc.Allrightsreserved.

    doi:10.1016/j.laa.2005.06.038

    304 C.T. Ng, W.Zhang/LinearAlgebra anditsApplications412(2006)303325

    operation arecalled iterativeequations . Many problemsofinvariantcurvescan be

    reducedtoiterativeequations[17].A basicform [3,12,13]ofaniterativeequationis

    T

    n

    (v) + λ n ? 1 T

    n ? 1

    (v) +???+ λ 1 T(v) + λ 0 v = F(v) ? v ? J, (1.1)

    where mapping F : J ? R on arealinterval J and constants λ j ? R are given.

    There are manyresultsonitscontinuous,differentiable and equivariantsolutions

    [10,16,22,25].Findingiterativeroots[12,13,21,23]ofaself-mapping F on J isto

    solvethespecialcase

    T

    n

    (v) = F(v) ? v ? J. (1.2)

    Afunction T : R ? R iscalled additive if T(u + v) = T(u) + T(v) forall u,v ? R .

    Our motivation hereisaboutthe additivesolutionsof(1.1). Asknown[1,p.13],

    additivefunctionson R arelinearovertherationals Q .Thusanadditivesolution of

    (1.1)isalinearoperator T over Q satisfying(1.1).

    Let V beavectorspaceovera?eld K ,andlet T bealinearoperatoron V .Inthis

    paper wesolvetheequation

    p(T) = F, (1.3)

    where p ? K [ x ] , deg (p) > 0,and F ,an algebraic operatoron V ,are given.This equation wassurveyedin[15,Chapter VIII].Resultson ?nite dimensional V over

    algebraicallyclosed K weregivenin[6,19].

    In Section 2 wegivesomepreliminaries.A canonicaldecompositiontheorem is statedthrough which we describethesolutionsof(1.3)forthespecialcase F = 0.

    Itisalso usedtoshow theexistenceofadditivesolutionsfor(1.1)on J = R when

    F = 0.

    InSection3acardinalsequence whichcharacterizesconjugacybetweenoperators isintroduced.Itisinsome waylinkedtothe Ulm invariants[14,pp.27,38,Theorem

    14]. Wegiveashortdiscussion on normalforms.Thenormalformscoverboththe Jordanandtherationalnormalformsasspecialcases.Itsinclusionisnotcrucialfor section 4,butit makesthepresentation ofexampleseasier.

    In Section 4,wegivethegeneralstepsinsolving(1.3),i.e. p(T) = F ,where F

    isalgebraic.Thespecialcase where F isnilpotentistreatedin Section5,yieldingan explicitlinearsystem which determinesthesolutionsbytheirsimilarityclasses.

    2. Preliminariesandtheequation p(T) = 0

    Let T be an operatoron a vectorspace V over K .For v/ = 0in V,Z(v,T) = Span { v,T(v),T

    2

    (v),... } denotesthe T -cyclicsubspace of V generated by v .The

    restriction of T to Z(v,T) ,denoted by T v ,is alinear operator on Z(v,T) . Ei-

    ther Z(v,T) hasin?nite dimension;elsethelowestpositiveinteger k suchthat

    { v,T(v),T

    2

    (v),...,T

    k

    (v) } islinearlydependentisitsdimension.Inthelattercase,

    say T

    k

    (v) =? λ k ? 1 T

    k ? 1

    (v) ????? λ 1 T(v) ? λ 0 v ,then m v (x) = x

    k

    + λ k ? 1 x

    k ? 1

    + 1 0 v

    n o T .

    L U b

    1 x

    n ? 1

    +? + λ 1 x + λ 0 ? K [ x ] .L B = ( 1 , 2 , n ) b a o

    f U t l o T o U d b

    j 1

    T( n ) =? λ 1 u n ???? λ 1 u 2 ? 0

asstsharacteristicolynomial.eferohisperator

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    I

    W i ie ve v o V c bw u a v =

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    w

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    eave

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    i ,s T

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