Table Of ContentSPRINGER BRIEFS IN MATHEMATICS
Daniel Alpay
Fabrizio Colombo
Irene Sabadini
Quaternionic de
Branges Spaces
and Characteristic
Operator Function
123
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Daniel Alpay Fabrizio Colombo
(cid:129) (cid:129)
Irene Sabadini
Quaternionic de Branges
Spaces and Characteristic
Operator Function
123
DanielAlpay Fabrizio Colombo
Schmid Collegeof Science andTechnology Dipartimento di Matematica
Chapman University Politecnico di Milano
Orange,CA, USA Milano,Italy
IreneSabadini
Dipartimento di Matematica
Politecnico di Milano
Milano,Italy
ISSN 2191-8198 ISSN 2191-8201 (electronic)
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Preface
This work inserts in the very fruitful study of quaternionic linear operators. This
study is a generalization of the complex case, but the noncommutative setting of
quaternions shows several interesting new features, see, e.g., the so-called S-
spectrum and S-resolvent operators. In this work, we study de Branges spaces,
namely, the quaternionic counterparts of spaces of analytic functions (in a suitable
sense) with some specific reproducing kernels, in the unit ball of quaternions or in
the half-space of quaternions with positive real parts. The spaces under consider-
ationwillbeHilbertorPontryaginorKreinspaces.Thesespacesarecloselyrelated
to operator models that are also discussed. The focus of this book is the notion of
characteristicoperatorfunctionofaboundedlinearoperatorAwithfiniterealpart,
andweaddressseveralquestionslikethestudyofJ-contractivefunctions,whereJ
is self-adjoint and unitary, and we also treat the inverse problem, namely, to
characterize which J-contractive functions are characteristic operator functions of
an operator. In particular, we prove the counterpart of Potapov’s factorization
theoreminthisframework.Besidesothertopics,weconsidercanonicaldifferential
equations in the setting of slice hyperholomorphic functions and we define the
losslessinversescatteringproblem.Wealsoconsidertheinversescatteringproblem
associated with canonical differential equations. These equations provide a con-
venient unifying framework to discuss a number of questions pertaining, for
example, to inverse scattering, non-linear partial differential equations and are
studied in the last section of this book.
A problem which is related to our study, which is one of the crucial problems in
operator theory, is determining the invariant subspaces of a linear closed operator.
WorkingonHilbertspaces,thespectraltheorem fornormaloperatorsisoneofthe
most important achievements of the last century. Even though there has been a lot
for works regarding the problem of extending the reduction theory to non-normal
linear operators, still a lot of problems are unsolved nowadays.
Inordertostudythespectralanalysisforanumberofnon-self-adjointoperatorsone
hastoextendthetheorytonon-normaloperatorsinaHilbertspaceandtooperators
inBanachspaces;thishasbeendonewiththetheoryofspectraloperators,see[65].
v
vi Preface
Animportantcontributiontothereductiontheoryofnon-normaloperatorswasthe
notion of characteristic operator function introduced by Livsic.
Inthequaternionicsettingthingsaremuchmorecomplicatedsincetheappropriate
notion,namely,theS-spectrumofaquaternioniclinearoperatorwasintroduced70
years after the paper of Birkhoff and von Neumann on the logic of quantum
mechanics that was published in 1936. Moreover, the spectral theorem for
quaternionic normal linear operators (bounded or unbounded) was proved in 2015
and appeared in the literature in 2016. We recall that, if T is a bounded linear
quaternionic operator then the S-spectrum is defined as
(cid:1)SðTÞ¼fs2H : T2(cid:2)2ReðsÞTþjsj2I is not invertibleg;
while the S-resolvent set is
‰SðTÞ:¼Hn(cid:1)SðTÞ:
LetusrestricttothatcasewhenT isaboundednormalquaternioniclinearoperator
on a quaternionic Hilbert space H. Then there exist three quaternionic linear
operatorsA,J,BsuchthatT ¼AþJB,whereAisself-adjointandBispositive,J
is an anti-self-adjoint partial isometry (called imaginary operator). Moreover, A, B
and J mutually commute.
LetussetCþ ¼fuþjv;ðu;vÞ2R(cid:3)Rþg,forj2S,whereSistheunitsphereof
j
purelyimaginaryquaternions.Sothespectral theorem isasfollows.Thereexists a
uniquespectralmeasureEj on(cid:1)SðTÞ\Cjþ sothatforanyslicecontinuousintrinsic
function f ¼f þ f j and x;y2H
0 1
Z Z
hfðTÞx;yi¼ f0ðqÞdhEjðqÞx;yiþ f1ðqÞdhJEjðqÞx;yi: ð1Þ
(cid:1)SðTÞ\Cjþ (cid:1)SðTÞ\Cjþ
With the spectral theorem and the S-functional calculus, that is the quaternionic
analogue of the Riesz-Dunford functional calculus, it turned out to be clear that to
replace complex spectral theory with quaternionic spectral theory we have to
replace the classical spectrum with the S-spectrum.
Thefirstdirectionofresearchofoperatortheoryinthequaternionicsetting,beyond
the spectral theorem based on the S-spectrum, was done in the recent long paper
[73], where the author studies quaternionic spectral operators. The present book
considers a different avenue and begins the investigation of the quaternionic
characteristic operator function. Inclassicaloperatortheorythenotion ofresolvent
operator plays a key role. More precisely, let T be a possibly unbounded linear
operatoractingonaHilbertspaceH.TheresolventoperatorRðzÞ¼ðT (cid:2)zI Þ(cid:2)1 is
H
an operator-valued function, analytic on the resolvent set ‰ðTÞ of T, assumed
non-empty, and its properties and those of T are closely related. Characteristic
operator functions are possibly simpler analytic functions, built on RðzÞ, and still
Preface vii
allowingtodeduce propertiesoftheoperatorfrom properties ofthefunctions. The
characteristic operator function associated with a close-to-unitary operator origi-
nates with the work of Livsic; see [93]. Properties and applications of the char-
acteristic operator function of an operator which is close-to-self-adjoint are
discussed in particular in the book [51].
This work consists of ten chapters, the Preliminaries being the first. Chapters 2–4
maybeseenofapreliminarynature,althoughtheyalsocontainsomenewmaterial.
In Chapter 2 we recall some facts on quaternions, quaternionic matrices and
quaternionic functional analysis. The main aspects needed in this work on the
theory of slice hyperholomorphic functions as well as on the S-resolvent operators
and the S-spectrum are recalled in Chapter 3. The original part in the section
consistsinthestudyofslicehyperholomorphicweights,bothinthecaseoftheunit
ball and of the half-space. A key tool is a map, denoted by x, which allows to
rewritethevaluesofaquaternionicvaluedfunctionintermsof2(cid:3)2matriceswith
complex entries.
The theory of slice hyperholomorphic rational functions and their symmetries is
considered in Chapter 4. Operator models in the sense of Rota are studied in
Chapter 5. In Chapter 6 we consider quaternionic HðA;BÞ spaces and we provide
the counterparts of various results in this framework, including the operator of
multiplication in the half-space case and in the unit ball case and the study of the
reproducing kernels.The case ofJ-contractive functions ispresented inChapter 7.
The characteristic operator function is defined and studied in Chapter 8, where we
alsoprovideexamplesandwediscussinverseproblems.Someclassesoffunctions
with a positive real part in the half-space or the unit ball are studied in Chapter 9.
Finally, Chapter 10 is devoted to the canonical differential systems in the quater-
nionicsetting,alsothoseassociatedwithanoperatorand,inparticular,westudythe
matrizant.
We consider this manuscript as a seminal work which can be expanded and can
give rise to several different directions of research in operator theory and hyper-
complex analysis.
Orange, USA Daniel Alpay
Milano, Italy Fabrizio Colombo
Milano, Italy Irene Sabadini
Contents
1 Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 The Complex Numbers Setting . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The Quaternionic Setting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Quaternions and Matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1 Quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Toeplitz and Hankel Matrices . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3 Some Remarks in Functional Analysis . . . . . . . . . . . . . . . . . . . 20
3 Slice Hyperholomorphic Functions . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.1 Slice Hyperholomorphic Functions. . . . . . . . . . . . . . . . . . . . . . 23
3.2 The S-Resolvent Operators and the S-Spectrum . . . . . . . . . . . . 27
3.3 The Map x and Applications . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.4 Slice Hyperholomorphic Weights: Half-Space
and Unit Ball Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4 Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.1 Rational Slice Hyperholomorphic Functions . . . . . . . . . . . . . . . 41
4.2 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5 Operator Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.1 Rota’s Model in the Quaternionic Setting. . . . . . . . . . . . . . . . . 47
5.2 Operator Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
6 Structure Theorems for HðA;BÞ Spaces. . . . . . . . . . . . . . . . . . . . . 53
6.1 HðA;BÞ Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
6.2 The Structure Theorem: Half-Space Case . . . . . . . . . . . . . . . . . 58
6.3 The Unit Ball Case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
6.4 The Conditions (6.15) and (6.25). . . . . . . . . . . . . . . . . . . . . . . 65
6.5 A Theorem on the Zeros of a Polynomial . . . . . . . . . . . . . . . . 67
ix
x Contents
7 J-Contractive Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
7.1 J-Contractive Functions in the Quaternionic Unit Ball . . . . . . . 69
7.2 J-Contractive Functions in the Right Half-Space. . . . . . . . . . . . 72
7.3 The Case of Entire Functions . . . . . . . . . . . . . . . . . . . . . . . . . 75
8 The Characteristic Operator Function . . . . . . . . . . . . . . . . . . . . . . 79
8.1 Properties of the Characteristic Operator Function. . . . . . . . . . . 79
8.2 Inverse Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
9 LðUÞ Spaces and Linear Fractional Transformations. . . . . . . . . . . 87
9.1 LðUÞ Spaces Associated with Analytic Weights . . . . . . . . . . . . 87
9.2 Linear Fractional Transformations and an Inverse Problem . . . . 90
10 Canonical Differential Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
10.1 The Matrizant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
10.2 The Characteristic Spectral Functions. . . . . . . . . . . . . . . . . . . . 101
10.3 Canonical Differential Systems Associated with
an Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
References.... .... .... .... ..... .... .... .... .... .... ..... .... 109
Index .... .... .... .... .... ..... .... .... .... .... .... ..... .... 115
