Canonical systems occupy a central position in the spectral theory of second order differential operators. They may be used to realize arbitrary spectral data, and the classical operators such as Schrödinger, Jacobi, Dirac, and Sturm-Liouville equations can be written in this form. ‘Spectral Theory of Canonical Systems’ offers a selfcontained and detailed introduction to this theory. Techniques to construct self-adjoint realizations in suitable Hilbert spaces, a modern treatment of de Branges spaces, and direct and inverse spectral problems are discussed. Contents Basic definitions Symmetric and self-adjoint relations Spectral representation Transfer matrices and de Branges spaces Inverse spectral theory Some applications The absolutely continuous spectrum

Theorems of factorising matrix functions and the operator identity method play an essential role in this book in constructing the spectral theory (direct and inverse problems) of canonical differential systems. Includes many varied applications of the general theory.

Canonical systems occupy a central position in the spectral theory of second order differential operators. They may be used to realize arbitrary spectral data, and the classical operators such as Schrödinger, Jacobi, Dirac, and Sturm-Liouville equations can be written in this form. ‘Spectral Theory of Canonical Systems’ offers a selfcontained and detailed introduction to this theory. Techniques to construct self-adjoint realizations in suitable Hilbert spaces, a modern treatment of de Branges spaces, and direct and inverse spectral problems are discussed. Contents Basic definitions Symmetric and self-adjoint relations Spectral representation Transfer matrices and de Branges spaces Inverse spectral theory Some applications The absolutely continuous spectrum

This book treats some basic topics in the spectral theory of dynamical systems, where by a dynamical system we mean a measure space on which a group of automorphisms acts preserving the sets of measure zero. The treatment is at a general level, but even here, two theorems which are not on the surface, one due to H. Helson and W. Parry and the other due to B. Host are presented. Moreover non­ singular automorphisms are considered and systems ofimprimitivity are discussed. and they are used to describe Riesz products, suitably generalised, are considered the spectral types and eigenvalues of rank one automorphisms. On the other hand topics such as spectral characterisations of various mixing conditions, which can be found in most texts on ergodic theory, and also the spectral theory of Gauss Dynamical Systems, which is very well presented in Cornfeld, Fomin, and Sinai's book on Ergodic Theory, are not treated in this book. A number of discussions and correspondence on email with El Abdalaoui El Houcein made possible the presentation of mixing rank one construction of D. S. Ornstein. Iam deeply indebted to G. R. Goodson. He has edited the book and suggested a number of corrections and improvements in both content and language.

The central topic of this book is the spectral theory of bounded and unbounded self-adjoint operators on Hilbert spaces. After introducing the necessary prerequisites in measure theory and functional analysis, the exposition focuses on operator theory and especially the structure of self-adjoint operators. These can be viewed as infinite-dimensional analogues of Hermitian matrices; the infinite-dimensional setting leads to a richer theory which goes beyond eigenvalues and eigenvectors and studies self-adjoint operators in the language of spectral measures and the Borel functional calculus. The main approach to spectral theory adopted in the book is to present it as the interplay between three main classes of objects: self-adjoint operators, their spectral measures, and Herglotz functions, which are complex analytic functions mapping the upper half-plane to itself. Self-adjoint operators include many important classes of recurrence and differential operators; the later part of this book is dedicated to two of the most studied classes, Jacobi operators and one-dimensional Schrödinger operators. This text is intended as a course textbook or for independent reading for graduate students and advanced undergraduates. Prerequisites are linear algebra, a first course in analysis including metric spaces, and for parts of the book, basic complex analysis. Necessary results from measure theory and from the theory of Banach and Hilbert spaces are presented in the first three chapters of the book. Each chapter concludes with a number of helpful exercises.

This textbook provides a graduate-level introduction to the spectral theory of linear operators on Banach and Hilbert spaces, guiding readers through key components of spectral theory and its applications in quantum physics. Based on their extensive teaching experience, the authors present topics in a progressive manner so that each chapter builds on the ones preceding. Researchers and students alike will also appreciate the exploration of more advanced applications and research perspectives presented near the end of the book. Beginning with a brief introduction to the relationship between spectral theory and quantum physics, the authors go on to explore unbounded operators, analyzing closed, adjoint, and self-adjoint operators. Next, the spectrum of a closed operator is defined and the fundamental properties of Fredholm operators are introduced. The authors then develop the Grushin method to execute the spectral analysis of compact operators. The chapters that follow are devoted to examining Hille-Yoshida and Stone theorems, the spectral analysis of self-adjoint operators, and trace-class and Hilbert-Schmidt operators. The final chapter opens the discussion to several selected applications. Throughout this textbook, detailed proofs are given, and the statements are illustrated by a number of well-chosen examples. At the end, an appendix about foundational functional analysis theorems is provided to help the uninitiated reader. A Guide to Spectral Theory: Applications and Exercises is intended for graduate students taking an introductory course in spectral theory or operator theory. A background in linear functional analysis and partial differential equations is assumed; basic knowledge of bounded linear operators is useful but not required. PhD students and researchers will also find this volume to be of interest, particularly the research directions provided in later chapters.

The focus program on Analytic Function Spaces and their Applications took place at Fields Institute from July 1st to December 31st, 2021. Hilbert spaces of analytic functions form one of the pillars of complex analysis. These spaces have a rich structure and for more than a century have been studied by many prominent mathematicians. They also have several essential applications in other fields of mathematics and engineering, e.g., robust control engineering, signal and image processing, and theory of communication. The most important Hilbert space of analytic functions is the Hardy class H2. However, its close cousins, e.g. the Bergman space A2, the Dirichlet space D, the model subspaces Kt, and the de Branges-Rovnyak spaces H(b), have also been the center of attention in the past two decades. Studying the Hilbert spaces of analytic functions and the operators acting on them, as well as their applications in other parts of mathematics or engineering were the main subjects of this program. During the program, the world leading experts on function spaces gathered and discussed the new achievements and future venues of research on analytic function spaces, their operators, and their applications in other domains. With more than 250 hours of lectures by prominent mathematicians, a wide variety of topics were covered. More explicitly, there were mini-courses and workshops on Hardy Spaces, Dirichlet Spaces, Bergman Spaces, Model Spaces, Interpolation and Sampling, Riesz Bases, Frames and Signal Processing, Bounded Mean Oscillation, de Branges-Rovnyak Spaces, Operators on Function Spaces, Truncated Toeplitz Operators, Blaschke Products and Inner Functions, Discrete and Continuous Semigroups of Composition Operators, The Corona Problem, Non-commutative Function Theory, Drury-Arveson Space, and Convergence of Scattering Data and Non-linear Fourier Transform. At the end of each week, there was a high profile colloquium talk on the current topic. The program also contained two semester-long advanced courses on Schramm Loewner Evolution and Lattice Models and Reproducing Kernel Hilbert Space of Analytic Functions. The current volume features a more detailed version of some of the talks presented during the program.