Table Of ContentOperator Theory: Advances and Applications
Vol. 209
Founded in 1979 by Israel Gohberg
Editors:
Joseph A. Ball (Blacksburg, VA, USA)
Harry Dym (Rehovot, Israel)
Marin us A. Kaashoek (Amsterdam, The Netherlands)
Heinz Langer (Vienna, Austria)
Christiane Tretter (Bern, Switzerland)
Associate Editors: Honorary and Advisory Editorial Board:
Vadim Adamyan (Odessa, Ukraine) Lewis A. Coburn (Buffalo, NY, USA)
Albrecht Bottcher (Chemnitz, Germany) Ciprian Foias (College Station, TX, USA)
B. Malcolm Brown (Cardiff, UK) J. William Helton (San Diego, CA, USA)
Raul Curto (Iowa City, lA, USA) Thomas Kailath (Stanford, CA, USA)
Fritz Gesztesy (Columbia, MO, USA) Peter Lancaster (Calgary, AB, Canada)
Pavel Kurasov (Lund, Sweden) Peter D. Lax (New York, NY, USA)
Leonid E. Lerer (Haifa, Israel) Donald Sarason (Berkeley, CA, USA)
Vern Paulsen (Houston, TX, USA) Bernd Silbermann (Chemnitz, Germany)
Mihai Putinar (Santa Barbara, CA, USA) Harold Wid om (Santa Cruz, CA, USA)
Leiba Rodman (Williamsburg, VI, USA)
llya M. Spitkovsky (Williamsburg, VI, USA)
Subseries
Linear Operators and Linear Systems
Subseries editors:
Daniel Alpay (Beer Sheva, Israel)
Birgit Jacob (Wuppertal, Germany)
Andre C.M. Ran (Amsterdam, The Netherlands)
Subseries
Advances in Partial Differential Equations
Subseries editors:
Bert-Wolfgang Schulze (Potsdam, Germany)
Michael Demuth (Clausthal, Germany)
Jerome A. Goldstein (Memphis, TN, USA)
Nobuyuki Tose (Yokohama, Japan)
lngo Witt (Gottingen, Germany)
Stability of
Operators and
Operator Semigroups
Tanja Eisner
Birkhauser
Author:
Tanja Eisner
Arbeitsbereich Funktionalanalysis
Mathematisches lnstitut
Auf der Morgenstelle 10
72076 TO bingen
Germany
e-mail: [email protected]
2010 Mathematics Subject Classification: 47 A1 0, 47D06, 47 A35, 28D05
Library of Congress Control Number: 2010930173
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ISBN 978-3-0346-0194-8
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Contents
Introduction 1
I Functional analytic tools 5
1 Compact semigroups . . . . . . . . . . . . 5
1.1 Preliminaries . . . . . . . . . . . . 5
1.2 Relatively compact sets in .Co-(X) . 7
1.3 Compact semitopological semigroups . 8
1.4 Abstract Jacobs-Glicksberg-de Leeuw decomposition for
operator semigroups . . . . . . . . . . . . . . . . 9
1.5 Operators with relatively weakly compact orbits 12
1.6 Jacobs-Glicksberg-de Leeuw decomposition for
Co-semigroups . . . . . 12
2 Mean ergodicity . . . . . . . . . . . . . . . 13
2.1 Mean ergodic operators ..... . 13
2.2 Uniformly mean ergodic operators 18
2.3 Mean ergodic Co-semigroups ... 20
2.4 Uniformly mean ergodic C -semigroups 22
0
3 Specific concepts from semigroup theory . . . . 23
3.1 Cogenerator . . . . . . . . . . . . . . . . 23
3.2 Inverse Laplace transform for C -semigroups 26
0
4 Positivity in .C(H) ............... . 32
4.1 Preliminaries . . . . . . . . . . . . . . . 32
4.2 Spectral properties of positive operators 33
4.3 Implemented operators . . 34
4.4 Implemented semigroups . 35
II Stability of linear operators 37
1 Power boundedness . . . . . . . . . . . 37
1.1 Preliminaries . . . . . . . . . . 37
1.2 Characterisation via resolvent . 40
1.3 Polynomial boundedness . . . . 43
vi Contents
2 Strong stability . . . . . . . . . . . . . . 45
2.1 Preliminaries .......... . 46
2.2 Representation as shift operators 48
2.3 Spectral conditions ...... . 49
2.4 Characterisation via resolvent . 53
3 Weak stability . . . . . . . . . . . . . 55
3.1 Preliminaries ......... . 55
3.2 Contractions on Hilbert spaces 57
3.3 Characterisation via resolvent . 60
4 Almost weak stability . . . 61
4.1 Characterisation .. 61
4.2 A concrete example 66
5 Category theorems . . . . 67
5.1 Unitary operators 67
5.2 Isometries .... . 71
5.3 Contractions .. . 72
6 Stability via Lyapunov's equation . 74
6.1 Uniform exponential and strong stability. 74
6.2 Power boundedness ............ . 76
III Stability of C0-semigroups 79
1 Boundedness . . . . . . . . . . . . . . . . 79
1.1 Preliminaries . . . . . . . . . . . 79
1.2 Representation as shift semigroups 81
1.3 Characterisation via resolvent . 82
1.4 Polynomial boundedness . 85
2 Uniform exponential stability ... 91
2.1 Spectral characterisation .. 91
2.2 Spectral mapping theorems 92
2.3 Theorem of Datko-Pazy . 94
2.4 Theorem of Gearhart. 96
3 Strong stability . . . . . . . . . . 99
3.1 Preliminaries ...... . 99
3.2 Representation as shift semigroups 102
3.3 Spectral conditions ...... . 103
3.4 Characterisation via resolvent . 105
4 Weak stability . . . . . . . . . . . . . 108
4.1 Preliminaries . . . . . . . . . . 108
4.2 Contraction semigroups on Hilbert spaces 109
4.3 Characterisation via resolvent . 112
5 Almost weak stability . . 114
5.1 Characterisation 114
5.2 Concrete example 120
Contents vii
6 Category theorems . . 122
6.1 Unitary case . 123
6.2 Isometric case . 126
6.3 Contractive case: remarks 127
7 Stability via Lyapunov's equation . 129
7.1 Uniform exponential stability 129
7.2 Strong stability . 130
7.3 Boundedness ........ . 132
IV Connections to ergodic and measure theory 133
1 Stability and the Rajchman property . . . . . . . . . . . . 133
1.1 From stability to the Rajchman property and back 133
1.2 Category result, discrete case .. 137
1.3 Category result, continuous case 141
2 Stability and mixing . . . . . . 141
2.1 Ergodicity ....... . 142
2.2 Strong and weak mixing 144
3 Rigidity phenomena . . . . . . 148
3.1 Preliminaries ..... . 148
3.2 Discrete case: powers of operators 150
3.3 Continuous case: Co-semigroups 156
3.4 Further remarks ........ . 161
V Discrete vs. continuous 163
1 Embedding operators into C -semigroups . . . . . 163
0
1.1 Sufficient conditions via functional calculus 164
1.2 A necessary condition ........... . 165
1.3 Normal operators and projections on Hilbert spaces 167
1.4 Isometries and co-isometries on Hilbert spaces 168
1.5 Abstract examples: a residuality result . 170
2 Cogenerators . . . . . . . . . . . . . . . . . . . . . . . 172
2.1 (Power) boundedness .............. . 172
2.2 Characterisation via cogenerators of the rescaled
semigroups . . . . . . . . 175
2.3 Examples ........ . 178
2.4 Polynomial boundedness . 180
2.5 Strong stability . . . . . . 181
2.6 Weak and almost weak stability . 182
Bibliography 183
List of Symbols 201
Index 203
Introduction
The real understanding involves, I believe, a synthesis of the discrete and
continuous ...
L. Lovasz, Discrete and Continuous: two sides of the same?
Evolving systems can be modelled using a discrete or a continuous time scale. The
n,
discrete model leads to a map 'P and its powers 'Pn on some state space while
the continuous model is given by a (semi)fiow ('Ptk;,o on n satisfying 'Po = Id
and 'Pt+s = 'Pt'Ps· In many situations, the state space n is a Banach space and the
maps are linear and bounded. In this case, we will use the notations T and T(t)
instead of 'P and 'Pt.
In this book, our focus is on the various concepts of stability of such linear
systems, where by stability we mean convergence to zero of {Tn}~=l or (T(t)k?.o
in a sense to be specified. This property is crucial for study of the qualitative be•
haviour of dynamical systems, even in the non-linear case. We adopt the general
philosophy of a parallel treatment of both the discrete and the continuous case,
systematically comparing methods and results. We try to give a reasonably com•
plete picture mentioning (most of) the relevant results, a strategy that has, by the
way, helped us to identify a number of natural open problems. However, we do not
discuss the case of positive operators and semigroups on Banach lattices, referring
the reader to, e.g., Nagel (ed.) [196] or the recent monograph of Emel'yanov [75]
for this topic. We also tried to minimise overlap with the monographs of van Neer•
ven [204] on asymptotics in the continuous case and Muller [191] in the discrete
case. Instead we emphasise the connections of stability in operator theory to its
analogues in ergodic theory and harmonic analysis.
In the following we summarise the content of the book.
Chapter I gives an overview on some functional analytic tools needed later.
Besides the classical decomposition theorems of Jacobs~Glicksberg~de Leeuw for
compact semigroups, we recall the powerful notion of the cogenerator of a Co•
semigroup. Finally, we present one of our main concepts for the investigation of
stability of C -semigroups, the Laplace inversion formula, which can also be seen
0
as an extension of the Dunford functional calculus for exponential functions.
2 Introduction
In Chapter II we investigate the behaviour of the powers of a bounded linear
operator on a Banach space. As a first step, we study power boundedness. Here and
later, behaviour of the resolvent of the operator near the unit circle plays a crucial
role. In particular, we give a resolvent characterisation for power boundedness on
Hilbert spaces using the £2-norm of the resolvent on circles with radius greater
than 1. We further study the related notion of polynomial boundedness which is,
surprisingly, easier to characterise than power boundedness.
Stability is the topic in the rest of the chapter. While uniform stability is
characterised by the spectral radius, we discuss strong stability and give various
characterisations of it, both classical and recent. When turning to weak stability,
we encounter a phenomenon, well-known in ergodic theory and described by Katok
and Hasselblatt [143, p. 748] as follows1
" ... It [mixing] is, however, one of those notions, that is easy and natural
to define but very difficult to study. .. ".
Thus very few results concerning weak stability are known, a sufficient condition
in terms of the resolvent being one of the exceptions.
We then introduce the concept of almost weak stability and give various
equivalent conditions. From this we see that almost weak stability, while looking
artificial, is much easier to characterise than weak stability. In particular, if the
operator has relatively compact orbits, then almost weak stability is equivalent to
"no point spectrum on the unit circle".
Almost weak stability turns out to be fundamentally different from weak
stability, as we see in Section 5. We show that a "typical" (in the sense of Baire)
unitary operator, a "typical" isometry, and a "typical" contraction on a separable
Hilbert space are almost weakly, but not weakly, stable. This is an operator the•
oretic counterpart to the classical theorems of Halmos [116] and Rohlin [221] on
weakly and strongly mixing transformations on a probability space.
We close the chapter by characterising stability and power boundedness of
operators on Hilbert spaces via Lyapunov equations.
In Chapter III we turn our attention to the time continuous case and consider
Co-semigroups (T(t) )t>O on Banach spaces. As in the previous chapter we discuss
boundedness and stability of (T(t) )t>o and try to characterise these properties by
the generator, its spectrum and resolvent. We start with boundedness and discuss
in particular the quite recent characterisation of a bounded Co-semigroup on a
Hilbert space by Gomilko [104] and Shi, Feng [231]. Their integrability condition
for the resolvent on vertical lines (Theorem 1.11) is the key to the resolvent ap•
proach to boundedness and stability, as first introduced by van Casteren [45, 46].
We then characterise generators of polynomially bounded semigroups (i.e., semi•
groups growing not faster than a polynomial) in terms of various resolvent condi•
tions.
1 Mixing corresponds to weak stability.
