Bergische Universität Wuppertal Doctoral Thesis General Extrapolation Spaces and Perturbations of Bi-Continuous Semigroups Author: Supervisor: Christian Budde Prof. Dr. Bálint Farkas Second Reader: Prof. Dr. Abdelaziz Rhandi A thesis submitted in fulfillment of the requirements for the degree of Doctor of Science (Dr. rer. nat.) in the Workgroup Functional Analysis School of Mathematics and Natural Sciences The PhD thesis can be quoted as follows: urn:nbn:de:hbz:468-20191205-110300-4 [http://nbn-resolving.de/urn/resolver.pl?urn=urn%3Anbn%3Ade%3Ahbz%3A468-20191205-110300-4] DOI: 10.25926/7ss7-bt33 [https://doi.org/10.25926/7ss7-bt33] Date of oral examination: 19. September 2019 Dean: Prof.Dr. Michael Günther Members of the examining board: Prof.Dr. Bálint Farkas Prof.Dr. Abdelaziz Rhandi Prof.Dr. Birgit Jacob Prof.Dr. Barbara Rüdiger-Mastandrea II Dedicated to my family and the love of my life III IV Acknowledgements First and foremost I would like to express my sincere gratitude to my advisor Prof.Dr. Bálint Farkas for the continuous support of my Ph.D study and related research, for his patience, motivation, and immense knowledge. It has been an honor to be his Ph.D student. His guidance helped me in all the time of research and writing of this thesis. I could not have imagined having a better advisor and mentor for my Ph.D study. The joy and enthusiasm he has for his research was contagious and motivational for me. I am also thankful for the excellent example he has provided as a successful mathematician and professor. My sincere thanks also goes to Prof.Dr. Abdelaziz Rhandi, Dr. Cristian Tacelli, Prof.Dr. Mar- jeta Kramar Fijavž, Dr. Retha Heyman and Dr. Ronalda Benjamin who provided me an opportunity to join their team as intern in Salerno (Italy), Ljubljana (Slovenia) and Stellen- bosch (South Africa). Without their precious support it would not be possible to reach my research goals. Especially, I want to thank the Team Outgoing of the International Office of the University Wuppertal which supported my research visits with the ERASMUS+ program. I am indebted to Prof.Dr. Moritz Weber und Viktoria Kühner for the additional proofreading of this thesis. I would like to thank my family for supporting me spiritually throughout writing this thesis and my life in general. Last but not least I would like to thank Lena, who was always there to help me. Christian Budde Bergische Universität Wuppertal July 2019 V VI Contents Acknowledgements V Introduction XI 1 Bi-Continuous Semigroups 1 1.1 Bi-admissible spaces . .1 1.1.1 The space Cb(Ω) of bounded continuous functions . .2 1.1.2 The dual space X0 .............................2 1.2 Bi-continuous semigroups . .3 1.2.1 The left-translation semigroup . .4 1.2.2 The adjoint semigroup . .4 1.3 The generator . .5 1.3.1 The left-translation semigroup on Cb(R) .................6 1.3.2 The adjoint semigroup . .7 1.4 The space of strong continuity . .8 1.4.1 The left-translation semigroup on Cb(R) .................8 1.4.2 The adjoint semigroup . .9 1.5 Well-posedness . .9 1.6 Adjoint bi-continuous semigroups . 10 1.7 Approximation theorems . 11 1.8 Notes . 12 2 Intermediate and Extrapolated Spaces 15 2.1 Sobolev and extrapolation spaces for invertible operators . 17 2.1.1 Abstract Sobolev spaces . 17 2.1.2 Extrapolation spaces . 20 2.2 Intermediate spaces for operators with rays of minimal growth . 25 2.3 Intermediate and extrapolation spaces for semigroup generators . 28 2.3.1 Intermezzo on real interpolation spaces . 31 2.4 Intermediate and extrapolation spaces for bi-continuous semigroups . 32 2.4.1 The Hille–Yosida Theorem . 32 2.4.2 Extrapolated semigroups . 34 2.4.3 Hölder spaces of bi-continuous semigroups . 36 2.4.4 Representation of Hölder spaces by generators . 41 2.5 Examples . 43 2.5.1 The left-translation semigroup . 43 VII 2.5.2 The multiplication semigroup . 45 2.5.3 The Gauss–Weierstrass semigroup . 46 2.6 Notes . 47 3 Fiberwise Multiplication 49 3.1 Lp-fiber spaces . 50 3.2 Unbounded operator-valued multiplication operators . 51 3.3 Extrapolation of unbounded multiplication operators . 53 3.4 Notes . 56 4 Desch–Schappacher Perturbations 57 4.1 An abstract Desch–Schappacher perturbation result . 59 4.1.1 Abstract Favard Spaces and comparison of semigroups . 62 4.2 Admissible operators . 63 4.3 Perturbations of the translation semigroup . 66 4.4 Notes . 71 5 Positive Miyadera–Voigt perturbations 73 5.1 Preliminaries . 74 5.1.1 Positivity and bi-AL-spaces . 74 5.1.2 Bounded perturbations . 78 5.1.3 Miyadera–Voigt perturbations . 78 5.2 The Proof . 81 5.3 Examples . 84 5.3.1 Rank-one perturbations . 84 5.3.2 Gauss–Weierstrass semigroup on M(R) .................. 85 5.4 Notes . 89 6 Implemented Semigroups 91 6.1 Preliminaries . 92 6.1.1 The space of bounded linear operators L (E) .............. 92 6.1.2 The implemented semigroup . 93 6.2 Intermediate and extrapolation spaces . 94 6.2.1 Favard- and Hölder spaces . 94 6.2.2 Extrapolation . 95 6.3 Perturbations . 96 6.3.1 Ideals in L (E) and module homomorphisms . 96 6.3.2 A one-to-one correspondence . 98 6.3.3 Comparisons . 100 6.4 Notes . 102 7 Flows on Networks 103 7.1 Preliminaries . 104 7.2 Transport problems on (in)finite metric graphs . 105 7.2.1 The simple case: constant velocities . 106 7.2.2 The rationally dependent case . 111 7.2.3 The general case for finite networks . 112 VIII 7.3 Notes . 114 Bibliography 115 IX X Introduction A lot of well-known partial differential equations modeling physical systems, such as the heat equation, the Schrödinger equation or the wave equation, use temporal change of states. Evo- lution equation is an umbrella term for such equations that can be interpreted as differential laws describing the development of a system or as a mathematical treatment of motion in time. In 1921 Albert Einstein said about physical models: As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality. Certainly, this quote encourages a discussion on the benefit of evolution equations in con- nection with the applications to physics, i.e., it exacerbates the discussion on the connection between mathematics and reality. However, one can say that the science provides models and we process and work them out. Hence, we are, as it seems, not responsible for the relation to reality. Of course, this is a short and crisp consideration of this topic, which normally requires much more discussion. This is for example done by G. Nickel [52, pp. 531-554]. Since this topic is beyond the goals of this thesis, we now leave this philosophic area and turn to mathematics. To quote Henri Poincaré: Mathematics has a threefold purpose. It must provide an instrument for the study of nature. But this is not all: it has a philosophical purpose, and, I daresay, an aesthetic purpose. With a solution of an evolution equation, one can predict the future of the corresponding physical system which makes it deterministic. One can find various books on the theory of evolution equations. At the same time, there are also monographs consisting only of mathematical applications of evolution equations in physics and life sciences, cf. [84], which in fact emphasizes the strength of the theory. Evolution equations can be treated by an operator theoretical approach. They can be rewritten as so-called abstract Cauchy problems. n We illustrate this by an example. Let Ω ⊆ R be a bounded domain with smooth boundary Pn ∂2 ∂Ω. Let ∆ := i=1 2 denote the Laplacian and consider the following problem ∂xi ∂ w(t, x) = ∆w(t, x), (t, x) ∈ [0, ∞) × Ω, ∂t w(0, x) = f(x), x ∈ Ω, (PDE) w(t, x) = 0, x ∈ ∂Ω, t ≥ 0, where f ∈ L2(Ω) is given. Now consider the Banach space X := L2(Ω) and define u(t) := w(t, ·) to be a function with variable x. Furthermore, define a linear operator by 2 Au := ∆u, D(A) := H0(Ω), XI 2 where H0(Ω) denotes the Sobolev space of functions with zero trace on the boundary, which ∞ is in fact the closure of C0 (Ω) with respect to the Sobolev norm. Observe that the boundary conditions in (PDE) are now incorporated into the domain of the operator, so that (PDE) can be rewritten as (u˙(t) = Au(t), t ≥ 0, (ACP) u(0) = x ∈ X, which is a Banach space valued initial value problem, also called an abstract Cauchy problem. The task is to analyse whether such an abstract Cauchy problem has a solution. Generally speaking, given an unbounded operator (A, D(A)) on a Banach space X, a (classical) solution of the corresponding abstract Cauchy problem (ACP) is by definition a function u : R≥0 → X such that u is continuously differentiable, u(t) ∈ D(A) for all t ≥ 0 and (ACP) is satisfied. In fact, this leads to the generic term well-posedness, including the existence of a unique solution of (ACP). By definition the equation is well-posed if D(A) is dense in X, for every x ∈ D(A) there exists a unique solution u(·, x) of (ACP) and for every sequence (xn)n∈N in D(A) with limn→∞ xn = 0 one has limn→∞ u(t, xn) = 0 uniformly for t on compact intervals [0, t0] for each t0 > 0. Now operator semigroups, the generalization of the exponential function, come into the pic- ture.