Spectral Properties and Companion Forms of Operator- and Matrix Polynomials [Elektronische Ressource] / Niels Vincentius Hartanto. Betreuer: Karl-Heinz Förster

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• Mathematics

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• Publié par: technische_universitat_berlin
• Publié le: 01 janvier 2011
• Nombre de lectures: 25
• Langue: English

Extrait

Spectral Properties and Companion Forms
of Operator- and Matrix Polynomials
vorgelegt von
Diplom-Mathematiker
Niels Vincentius Hartanto
Chemnitz
von der Fakult¨at II - Mathematik und Naturwissenschaften
der Technischen Universit¨at Berlin
zur Erlangung des akademischen Grades
Doktor der Naturwissenschaften
- Dr. rer. nat. -
genehmigte Dissertation
Promotionsausschuss:
Vorsitzender : Prof. Dr. Wolfgang K¨onig
Berichter/Gutachter: Prof. Dr. Karl-Heinz F¨orster
Berichhter: Prof. Dr. Volker Mehrmann
Berichter/Gutachter: Prof. Dr. B´ela Nagy
Tag der wissenschaftlichen Aussprache: 04. Mai 2011
Berlin 2011
D83”Cruelty to animals is one of the most significant vices of a low and ignoble people.
Wherever one notices them, they constitute a sure sign of ignorance and brutality
which cannot be painted over even by all the evidence of wealth and luxury. Cruelty
to animals cannot exist together with true education and true learning.”
– Alexander von Humboldt
†
”I am not interested to know whether vivisection produces results that are profitable
to the human race. The pain which it inflicts upon unconsenting animals is the basis
of my enmity toward it, and it is to me sufficient justification of the enmity without
looking further”.
– Mark Twain
†
”Imagine living your life in a small, filthy cage constantly in pain, unable to stand
or lie down comfortably. After months of agony, your torture finally ends, but not
at the slaughterhouse. Instead, two gentle hands reach down to lift you out of the
darkness, and bring you to a safe, loving place. For the first time in your life you
can stretch your wings and legs and feel soft straw and cool grass beneath your feet.”
– Dr. Karen Davis
†Acknowledgements
Die Fertigstellung dieser Arbeit w¨are nicht m¨oglich gewesen ohne die Hilfe und
Unterstutzung¨ einiger Menschen, die ich hier dankend erw¨ahnen m¨ochte.
Zun¨achst m¨ochte ich mich ganz herzlich bei Karl-Heinz F¨orster bedanken. Das
Thema der Dissertation, dessen Richtung er vorgegeben hat, war spannend und
bereitete mir viel Freude. Besonders dankbar bin ich fur¨ die stetige und geduldige
Unterstutzung¨ in Form von vielen langen Diskussionen, hilfreichen Anregungen und
konstruktiver Kritik und fur¨ die vielen motivierenden Impulse.
Zu besonders großem Dank, nicht nur fur¨ die Begutachtung dieser Dissertation,
bin ich Volker Mehrmann verpflichtet. Seine anf¨anglichen Vorschl¨age und Anregun-
gen haben die sp¨atere Richtung der Arbeit wesentlich beeinflusst. Die fachlichen
Gespr¨ache mit ihm waren stets enorm effizient und hilfreich und seine Kritik in
Bezug auf das Verfassen von wissenschaftlichen Arbeiten sind fur¨ mich von sehr
hohem Wert gewesen.
Sehr dankbar bin ich auch Christian Mehl und Michael Karow fur¨ die hilfreichen
Gespr¨ache und Diskussionen und Bela Nagy fur¨ die Begutachtung dieser Arbeit.
I would like to thank Dario Bini very much for active and invaluable email discus-
sions and Federico Poloni for the inspiring exchange.
Meiner Familie m¨ochte ich großen Dank fur¨ die private Unterstutzung¨ aussprechen.
Ohne sie h¨atte diese Arbeit nicht entstehen k¨onnen. Magda Nafalska danke ich sehr
fur¨ das Korrekturlesen und fur¨ ihre stets offenen Ohren und motivierenden Worte
in etwas schwierigeren Zeiten.Contents
Introduction 8
1 Spectral properties of algebra valued functions that are analytic in
an annulus 13
1.1 Introduction and general results . . . . . . . . . . . . . . . . . . . . . 13
1.2 Matrix valued functions . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.3 Matrix polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2 m–monic operator functions which are analytic on an annulus with
self adjoint coefficients 31
2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2 Rotation invariance of eigenvalues . . . . . . . . . . . . . . . . . . . 33
2.3 m–monic operator polynomials with Hermitian coefficients . . . . . . 40
3 Degreereductionof m–monicmatrixpolynomialsandpreservation
of spectral properties 41
3.1 Degree reduction of polynomials with coefficients that are elements
of a complex Banach algebra . . . . . . . . . . . . . . . . . . . . . . 41
3.2 Reduction and factorizations . . . . . . . . . . . . . . . . . . . . . . 45
3.3 Recovery of Jordan chains . . . . . . . . . . . . . . . . . . . . . . . . 50
4 m–monic matrix polynomials with entrywise nonnegative coeffi-
cients 53
4.1 Factorizations of 1–monic matrix polynomials . . . . . . . . . . . . . 54
4.2 F of m–monic matrix p . . . . . . . . . . . . 60
4.3 Nonnegative irreducible matrix polynomials . . . . . . . . . . . . . . 64
∑l j4.4 The operator equation X = A X . . . . . . . . . . . . . . . . 86jj=0
5 Computing spectral factorizations of m–monic matrix polynomials
with a cyclic reduction algorithm 88
5.1 Transformation to a Markov problem . . . . . . . . . . . . . . . . . . 91
5.2 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6 Conclusions 99
References 101
7Introduction
Let m∈N, m> 1 and G⊂C be a domain and letA be a complex Banach algebra.
We consider functions F :G→A of the type
mF(λ) =λ I−A(λ), (I)
∑
jwhere I denotes the identity in A and A(λ) = λ A , A ∈ A, is a Laurentj jj∈Z
series. Wewillcallfunctionsofthistypem–monicfunctions. Inapplicationsvarious
special cases of of this type appear.
For instance, in modeling of queueing problems, where Markov chains play a major
role, matrix functions of the type
∞
∑
j+1F(λ) =λI − λ Aj
j=−1
with entrywise nonnegative n×n matrix coefficients A arise. They are associatedj
withthetransitionmatrixoftheMarkovchaintheentriesofwhicharethetransition
probabilities from one to each other state of the stochastic process. The coefficients
∑
∞ nsatisfy the condition ( A )1 = 1 , where 1 denotes the vector in R thej n n nj=−1
entries of which are all equal to 1. Very crucial for the analysis of these Markov
chains is the minimal entrywise nonnegative solution of the matrix equation
∞
∑
j+1X = A X .j
j=−1
It is strongly connected to certain factorizations of the function F and key to find
so-called stationary vectors, which represent equilibrium probability distributions
oftheMarkovchain. Seee.g. [BLM05],[GHT96],[LR99],[Mei06],[Neu94],[Neu89].
Another field of application is in hydrodynamics. For instance, the study of small
motions and normal oscillations of a viscous incompressible fluid in an open con-
tainer leads to a spectral problem for functions of the type
−1L(λ) =I−λA −λ B,
where A is a positive definite and B a nonnegative definite operator. See e.g.
[AKL68], [AHKM03], [KL68]. Multiplying with λ on both sides leads to the 1–
monic function
2F(λ) =λL (λ) =λI −(λ A+B).
Another example can be found in [AKS97], [Suk97], where the investigation of
small convective motions of a fluid in a container results in spectral problem for the
operator function
2 2L (λ) =λ A−λ(ϵQ −I)+C =λI −(−λ A+λϵQ −C),ϵ
where A,Q,C are compact self adjoint operators in a Hilbert space, A is positive
definite, C is nonnegative definite and ϵ is a positive parameter.
Another instance where m–monic functions arise implicitly is in [Mar88], where
factorizations of functions with arbitrary operator coefficients are investigated.
8However, m–monic functions have not been studied explicitly, yet. Clearly, for any
arbitrarily chosen m∈N, any function in Laurent representation can be rewritten
as an m–monic function. Nevertheless, the m–monic form loses this arbitrariness
when the coefficients of A are supposed to have special properties like entrywise
nonnegativity, nonnegative definiteness.
The intention of this thesis is to study spectral properties of m–monic functions
where the coefficients A are elements of a complex Banach algebra A. A majorj
focus lies on the two cases where the coefficients are linear bounded self adjoint
operators in a Hilbert space or entrywise nonnegativen×n-matrices. Furthermore,
we study very closely the case when F is a polynomial.
ThebehaviorofthespectralradiusofA(λ)asλvariesthroughGhasaconsiderable
impact on the structure and distribution of the spectral points of F. We exploit
this connection by investigating the real valued function
+ϕ : G∩[0,∞)→R , ϕ (τ) = maxsprA(λ),A A
|λ|=τ
where sprA(λ) denotes the spectral radius of A(λ).
This strategy has already been used for special cases in [FN05a] to study spectral
properties of quadratic 1– monic matrix polynomials where the coefficients of A are
entrywise nonnegative matrices and the sum of the coefficients is irreducible. Fac-
torizations of m–monic polynomials are studied in [FN05b], where the coefficients
are elements of a cone in an ordered Banach algebra.
A similar approach has been used by H. K. Wimmer and J. Swoboda. The pe-
ripheral eigenvalues of monic matrix polynomials (monic polynomials of degree m
are m–monic) with Hermitian coefficients are investigated in [Wim08] under a con-
.dition which is closely related to the spectral radius of |A|(λ), where |A|(λ) .=
∑
1j 2 /2λ (A ) . [SW10] studies the spectrum of monic operator polynomials withjj∈Z
bounded nonnegati