Limit operators and applications on the space of essentially bounded functions [Elektronische Ressource] / vorgelegt von Marko Lindner
100 Pages
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Limit operators and applications on the space of essentially bounded functions [Elektronische Ressource] / vorgelegt von Marko Lindner

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100 Pages
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Limit Operators and Applicationson the Spaceof Essentially Bounded Functionsvon der Fakultat¨ fur¨ Mathematikder Technischen Universitat¨ ChemnitzgenehmigteDissertationzur Erlangung des akademischen GradesDoctor rerum naturalium(Dr. rer. nat.)vorgelegt vonDiplom-Mathematiker Marko Lindner,geboren am 20. Oktober 1973 in Zschopau,eingereicht am 7. April 2003.Gutachter: Prof. Dr. B. SilbermannProf. Dr. S. N. Chandler-Wilde¨Prof. Dr. W. SprossigTag der Verteidigung: 8. Dezember 2003IntroductionThis thesis is concerned with the description of the behaviour at infinity of so called∞ ∞ nband-dominated operators on L :=L (R ). We are especially interested in a prop-erty called invertibility at infinity which turns out to be the key to some other inter-esting features of the operator under consideration, including Fredholmness and theapplicability of several approximation methods. We will discuss and prove all thesecross connections.Our main tool for the investigation of invertibility at infinity are limit operators.Forthepracticabilityofthederivedcriteriaitisessentialtohavesomegoodknowledgeof limit operators; and on the journey through this subject we prove some previouslyunknown facts which – by the way – are beautiful and interesting on their own. Thisis why parts of this thesis may be understood as an approach to the understanding oflimit operators – a process that certainly still needs some time.

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Limit Operators and Applications
on the Space
of Essentially Bounded Functions
von der Fakultat¨ fur¨ Mathematik
der Technischen Universitat¨ Chemnitz
genehmigte
Dissertation
zur Erlangung des akademischen Grades
Doctor rerum naturalium
(Dr. rer. nat.)
vorgelegt von
Diplom-Mathematiker Marko Lindner,
geboren am 20. Oktober 1973 in Zschopau,
eingereicht am 7. April 2003.
Gutachter: Prof. Dr. B. Silbermann
Prof. Dr. S. N. Chandler-Wilde
¨Prof. Dr. W. Sprossig
Tag der Verteidigung: 8. Dezember 2003Introduction
This thesis is concerned with the description of the behaviour at infinity of so called
∞ ∞ nband-dominated operators on L :=L (R ). We are especially interested in a prop-
erty called invertibility at infinity which turns out to be the key to some other inter-
esting features of the operator under consideration, including Fredholmness and the
applicability of several approximation methods. We will discuss and prove all these
cross connections.
Our main tool for the investigation of invertibility at infinity are limit operators.
Forthepracticabilityofthederivedcriteriaitisessentialtohavesomegoodknowledge
of limit operators; and on the journey through this subject we prove some previously
unknown facts which – by the way – are beautiful and interesting on their own. This
is why parts of this thesis may be understood as an approach to the understanding of
limit operators – a process that certainly still needs some time.
We will start by describing the already mentioned main items in some more detail.
Band- and band-dominated operators
∞An operator on‘ (Z) is regarded as a band operator if its matrix representation with
respect to the canonical basis is a band matrix – a matrix whose entries a vanish ifij
|i−j| exceeds some number which is then called the band-width.
∞ nBy an analogous construction, band operators are defined on‘ (Z ). The passage
from the discrete case to the function case is done by a rather natural identification
nwhich essentially is nothing but “cutting” R into small hypercubes: Every function
∞ ∞ n ∞ nf ∈L can be identified with anL [0,1) -valued‘ (Z ) sequence, and by this iden-
∞ ∞ n ∞ ntification, an operator on L can be identified with an operator on ‘ (Z ,L [0,1) ).
∞Thus, we can carry over the notion of band operators to operators on L .
∞ ∞Clearly, the set BO of all band operators on L is a subalgebra ofL(L ) – but it
is not closed. Therefore one introduces the closure of BO with respect to the operator
norm, which shall be denoted by BDO, and its elements are referred to as band-
dominated operators.
Band- and band-dominated operators reside in many fields of mathematics and
physics. Just to mention a few examples, we find them in acoustic scattering and
propagation problems [6, 7, 8, 9], quantum mechanics [17], small-world networks [28]
and,ofcourse,asdiscretizationsofpartialdifferentialandpseudo-differentialoperators.
34 INTRODUCTION
The study of concrete classes of band and band-dominated operators (such as con-
volution , Wiener-Hopf , and Toeplitz operators) goes back to the 1930’s starting with
[40] by Wiener and Hopf, continued in the 1950’s, for example, in [19] and [15] by
Gohberg and Krein, and was culminating in the 1970/80’s with the huge mono-
¨graphs [14] byGohberg/Feldman and [4] byBottcher/Silbermann. A general
theory for this class of operators was initiated by Simonenko [36], [37] whose con-
structions (after some appropriate modifications) can still be found in the proof of
Proposition 3.3, an essential ingredient to the proof of our Theorem 1.
Invertibility at infinity
An operator A is said to be invertible at infinity if there are two operators B ,B and1 2
na bounded and measurable subset U ofR such that
Q AB = Q = B AQU 1 U 2 U
holds, where Q is the operator of multiplication by the function that is 0 on and 1U
outside U. For band-dominated operators there is an equivalent description of this
property, given in Definition 1.28, which is more convenient for us.
This property is closely related with Fredholmness. Indeed, in the discrete cases
p p‘ , 1 < p < ∞, both properties even coincide. In the cases L , 1 < p < ∞, invert-
ibility at infinity is still necessary for Fredholmness. In Chapter 6 we will show that
conversely, invertibility at infinity plus some other – readily verifiable – property is
sufficient for Fredholmness in an algebra generated by convolution and multiplication
operators which is containing many interesting cases. So by following one of these two
concepts,wewillnotcompletelylosecontacttotheotherone,andwewilldemonstrate,
why it is convenient for us to follow the notion of invertibility at infinity rather than
Fredholmness.
Every convolution operator with an absolutely summable kernel function is band-
dominated. Moreover, every operator in the algebra generated by such convolution
operators and bounded multiplication operators is band-dominated. So we will be
able to give a criterion on invertibility at infinity for such operators, provided the
multiplication operators involved do not behave “too strangely” at infinity. We will
specify clearly what kind of behaviour we do expect there, but it is important to note
that we do not need convergence at infinity, as many of our fore-runners did. The
reason for that is the special tool we are using:
Limit operators
In many situations one has to describe an operator’s behaviour at some singular, out-
standing point θ. This is where limit operators enter the scene. What is probably the
first application goes back to the late 1920’s, where Favard [13] used them to study
ordinary differential equations with almost-periodic coefficients.INTRODUCTION 5
Since that time limit operators have been used in the context of partial differential
and pseudo-differential operators and in many other fields of numerical analysis (for
instance, see [16]). The first time limit operator techniques were applied to the general
case of band-dominated operators was in 1985 by Lange and Rabinovich [21].
p pTypical applications lead to the study of the behaviour of operators on ‘ or L
at the point θ = ∞. For instance, Lange/Rabinovich, Rabinovich and Rabi-
novich/Roch/Silbermann study Fredholmness in [21], [22], [23], [30], [31], [32], the
latter three and the author study invertibility at infinity in [33], [25] and the applica-
bility of approximation methods in [32], [33], [26].
¨In a somewhat different situation, Bottcher, Karlovich and Rabinovich in
[3] use limit operators to study the local behaviour of singular integral operators at the
endpoint θ of a Carleson curve in the plane. We will here focus our investigations on
the case p =∞ and θ =∞:
∞ nTake an arbitrary operator A on L . Given a point θ ∈R , one might ask what
A “looks like” from another “point of view”, for instance, the point θ taking the role
nof the origin inR . Well, for finite points θ, the answer is just V AV (where V is a−θ θ θ
∞ nsimple shift operator acting on L by (V u)(x+θ) = u(x) ∀x ∈R ) which is still aθ
very close relative of the operator A itself.
But what if we ask for the point θ = ∞? Then θ can only be approached as the
limit of a sequence h of finite points h , m = 1,2,..., and by doing the above processm
for every one of these finite points h , the answer for θ can only be understood inm
the sense of some sort of limit – for instance, a so called P−limit – of the sequence
V AV , m = 1,2,... as m goes to infinity. And this is what we will call a limit−h hm m
operator. We will denote this limit operator by A since it heavily depends on theh
choice of the sequence h = (h ).m
The behaviour ofA atθ =∞ can be expressed by a whole bunch of limit operators
{A }. We collect all of them in the so called operator spectrum which is denoted byh
op ∞σ (A). If a band-dominated operator A on L has sufficiently many limit operators
in the sense that every sequence tending to infinity has an infinite subsequenceg such
thatA exists – we writeA∈B and regardA as a rich operator in this case – we cang $
prove the following theorem.
Theorem 1 An operator A ∈ B is invertible at infinity if and only if its operator$
opspectrum σ (A) is uniformly invertible.
...where “uniformly invertible” means
opa) all elements of σ (A) are invertible and
b) their inverses are collectively bounded.
This theorem links the property of invertibility at infinity to the study of limit
operators. During the last years Rabinovich, Roch and Silbermann proved it in
p 2the discrete cases ‘ with 1 < p <∞ [32] and in the case L [33]. Our investigations
are heavily inspired by these two papers and we demonstrate in the example of the
∞ 1 ∞ pmost unloved case, L , how the remaining cases ‘ , ‘ and L , p = 2 have to be
treated. In either case, Theorem 1 proves to be valid.
66 INTRODUCTION
In our case, p = ∞, we then tackle what has grown to some sort of “the big
question” in limit operator business, namely: Can the word “uniformly” be replaced
1by “elementwise” in Theorem 1? In other words, may we drop condition b) ?
As a first serious step towards finding the answer, we prove that, as true for almost
all sets named “spectrum”, operator spectra of rich operators enjoy some sort of a
compactness property. In detail, we are able to prove that the operator spectrum
of a rich operator is sequentially compact with respect to P−convergence. This is a
first indication that the answer might be “Yes” – the more beautiful but less expected
answer!
Employing this compactness property, we will indeed succeed in proving that
Theorem 2 The (global as well as every local) operator spectrum of a rich operator
A∈B is automatically uniformly invertible, provided it is elementwise invertible.$
1 ∞ 1 ∞...hereby giving an answer to the “big question” in the cases ‘ , ‘ , L and L ,
namely: “Yes!” Now we clearly have to refine Theorem 1, which is of course worth
one more theorem.
Theorem 3 An operator A∈B is invertible at infinity if and only if all of its limit$
operators are invertible.
Beingtheconglomerationoftheprecedingtheorems, thisisamajoroutcomeofthis
thesis. Moreover, it will be our starting point for the study of invertibility at infinity
and all of its applications. In order to really work with Theorem 3, one still has to gain
some knowledge on the objects it is dealing with:
1. What are the elements ofB ? and$
2. What do their limit operators look like?
∞1.) B turns out to be a Banach subalgebra of L(L ), and we establish some$
of its ingredients, including convolution operators and operators of multiplication by:
uniformly continuous functions, slowly oscillating functions, periodic functions, step
functions and, of course, convergent functions. Moreover, by Proposition 2.9 b), we
prove one more interesting fact about the structure of B , thereby giving a positive$$
answer to another open question stated by Rabinovich, Roch and Silbermann.
2.) In many applications one has to deal with operators A that are a sum of
products of convolution and multiplication operators. Since convolution operators are
translation invariant, all of their limit operators coincide with the operator itself. So
it remains to study limit operators of multiplication operators in some more detail:
∞ ∞By L we denote the set of all b∈L for which the multiplication operator M isb$
rich. Studying different classes of functions b we find and prove
∞Theorem 4 BUC = BC∩L
$
where BC and BUC denote, respectively, the class of bounded and continuous
nfunctions and the class of bounded and uniformly continuous functions onR .
1which is also referred to as “the nasty condition” by some authorsINTRODUCTION 7
Moreover, we can show that
∞Theorem 5 SO = CL∩L
$
holds for the class SO of slowly oscillating functions, where CL is the set of all
∞b∈L such that all limit operators of M are multiples of the identity operator I.b
Intuition, confirmed by Theorems 3 and 5, indicates that limit operatorsA cannoth
be any more complicated – in many cases they are indeed “simpler” – than A itself.
We will discuss this fact and will show that this process of “simplification” is ultimate
– in the sense that things are not going to become even simpler when doing this step
opagain – and hence, one can think ofσ (A) as a collection ofA’s most elementary parts
– atoms or whatsoever – which cannot be split any more, at least not by the same
instrument. Note that the results presented in this discussion about “simplification”
are not new, but writing down these things in such context was never done before
although it seems to be important for the understanding of limit operators and of our
results.
Approximation methods
If one has to solve an equation of the form
Au = b, (1)
∞ ∞ ∞where A∈L(L ) and b∈ L are given, and u∈ L is to be determined, one often
tries to approximate the operator A by a sequence of operators (A ) in order toτ τ>0
solve the somewhat easier equations
A u = b (2)τ τ
instead, hoping these are uniquely solvable – at least for all sufficiently large τ – and
thatthesolutionsu of(2)insomesensetendtoasolutionuof(1)asτ goestoinfinity.τ
∞If this is the case for every right-hand sideb∈L , then we say that the approximation
method (A ) is applicable to A.τ
In many cases (for strong convergence A → A this is Polski’s theorem, andτ
in the case we have in mind this was done by Roch and Silbermann in [34]) the
applicability of (A ) to A can be shown to be equivalent to A being invertible plusτ
(A ) being stable, the latter means that all A are invertible for sufficiently large τ,τ τ
and their inverses are collectively bounded.
So the essential question is that about the stability of a given sequence of operators
∞ ∞ nA , each of them acting on L = L (R ). We find the answer to that question byτ
(temporarily) increasing the dimension of the problem, which goes in the following
way:
n+1 nClearly, R is what results from stacking a continuum of copies of the R into
the (n+1)-th dimension. By the same idea, stacking the operators A (put A := Iτ τ
∞ n+1forτ ≤ 0) into the (n+1)-th dimension results in an operator onL (R ) which we
will denote by⊕A .τ8 INTRODUCTION
τ τ

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6 6
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The finite section sequence (A ) and its stacked operator⊕A for n = 1.τ τ>0 τ
We will introduce the notion of sufficiently smooth sequences, showing that, under
fairly general assumptions on the smoothness of the mapping τ 7→ A , the sequenceτ
(A ) is stable if and only if ⊕A is invertible at infinity! Now Theorem 3 is involved,τ τ
opgiving us a criterion for this stability in terms of elementwise invertibility ofσ (⊕A ).τ
If (A ) is the so called finite section method for A, then we have a very clearτ
description of all limit operators of ⊕A which also allows us to pass back to the n-τ
dimensional context and to operators that are related toA itself and to limit operators
of A. These investigations culminate in Theorem 5.2 on the finite section method.
pThis approach was successfully applied to the discrete cases ‘ , 1<p<∞, in [32]
2and to L in [33] byRabinovich, Roch andSilbermann. The stacking idea works
p pfine and without any problems in ‘ , while things become much more delicate in L ,
when stacking a continuum of operators. The method to cope with these difficulties
presented in [33] is connected with some rather drastic restrictions to the operator.
We will demonstrate a much more natural idea here, by studying what we will call the
essential invertibility of sufficiently smooth sequences.
In Chapter 6 we will apply our results on approximation methods to some concrete
operators – and algebras generated by them – culminating in an outlook to problems
of propagation and scattering of (for example, acoustic) waves.
∞ pL versus L
∞As already mentioned, our studies are mainly concentrated on the space L . In some
psense this is the most outstanding and challenging case of L which often needs some
extra treatment.
∞ 1 1 0Firstly, L is not reflexive. Its dual space is strictly larger than L , while (L )
∞ p p 0 qequalsL . In all other spacesL , 1<p<∞, one has (L ) =L with 1/p+1/q = 1.
∞So inL , the identification of adjoint operators and the verification of their properties
paremuchmoresophisticatedthaninL . InSection1.2.3wewillshowhowtoovercome
these difficulties.INTRODUCTION 9
n nSecondly, for the hypercube C := [−1,1] ⊂R the operator of multiplication by
thecharacteristicfunctionofthesetmC stronglyconvergestotheidentityoperatoron
pL as m→∞ if and only if p<∞. This makes the study of approximation methods
∞ pin L a somewhat more delicate problem than in L , as can be seen, for instance, in
∞ pSection 4.3.3. Other consequences of this discrepancy betweenL andL are different
relations between the idealsJ andK introduced in Section 1.4.
∞Fortunately, there is also one aspect which sometimes makes life a bit easier inL
p ∞than in L : A L -function (almost) attains its norm on an arbitrarily small subset of
n
R . This fact is used, for instance, in the proof of Theorem 2. Another place, where
we derive benefit from the properties of the supremum norm, is the proof of Lemma
1.8. But, while Lemma 1.8 essentially can be replaced by some proposition from [32]
pin the case ofL , we have not yet succeeded in transferring the proof of Theorem 2 to
pL .
Remark on publications
Essential parts of this thesis have already been (or are currently about to be) made
accessible to the scientific community in the articles [24] and [25], co-authored by
Bernd Silbermann, as well as in [26] and [27].
Acknowledgements
I am grateful to my advisors and friends Bernd Silbermann and Steffen Roch
for countless fruitful conversations, inspirations and hints. Moreover, I want to thank
Simon Chandler-Wilde for many nice conversations and for sharing his knowledge
with me, and Thomas Gruhle for helping me with this splendid cover page.
Finally, and on top of all, I am thankful to somebody very special for being my
muse, holding my hand and believing in me all the time: Danke,Diana!Contents
Introduction 3
1 Preliminaries 13
1.1 Basic agreements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.1.1 Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.1.2 Banach spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.1.3 Argument domains . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.1.4 Functions and sequences . . . . . . . . . . . . . . . . . . . . . . 14
1.1.5 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.1.6 The system case . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.1.7 Agreement on Banach algebras . . . . . . . . . . . . . . . . . . 15
1.2 Classes of operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.2.1 Operators of shift and multiplication . . . . . . . . . . . . . . . 15
1.2.2 Band- and band-dominated operators . . . . . . . . . . . . . . . 16
1.2.3 The algebraS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.2.4 The algebrasB andB . . . . . . . . . . . . . . . . . . . . . . . 22S
1.3 Invertibility of sets of operators . . . . . . . . . . . . . . . . . . . . . . 24
1.4 Invertibility at infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.5 Approximation methods . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.5.1 Motivation and Definition . . . . . . . . . . . . . . . . . . . . . 29
1.5.2 Additional approximation methods . . . . . . . . . . . . . . . . 29
1.5.3 Which type of convergence is appropriate? . . . . . . . . . . . . 30
1.6 P−convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.6.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.6.2 Characterization and Properties . . . . . . . . . . . . . . . . . . 31
1.7 Applicability vs. Stability . . . . . . . . . . . . . . . . . . . . . . . . . 34
10