This book offers a comprehensive exploration of representation theorems, presenting direct proofs for two distinct classes of Hardy spaces. It delves into the theoretical underpinnings and practical implications of these theorems, making complex concepts accessible for readers. The clear explanations and structured approach aim to enhance understanding of Hardy spaces and their applications in mathematical analysis.
Focusing on operator theory, this textbook is designed for graduate students and offers a clear introduction to the subject. It features twenty examples of operators, exploring key concepts such as norm, spectrum, commutant, and invariant subspaces. Each operator is examined in detail, highlighting its unique properties and applications, making it a valuable resource for those looking to deepen their understanding of this mathematical field.
This textbook serves as an accessible introduction to a complex field, making it suitable for newcomers. It not only covers foundational concepts but also highlights potential areas for future research, encouraging further exploration and study.
Inner functions form an important subclass of bounded analytic functions. Since they have unimodular boundary values, they appear in many extremal problems of complex analysis. They have been extensively studied since early last century, and the literature on this topic is vast. Therefore, this book is devoted to a concise study of derivatives of these objects, and confined to treating the integral means of derivatives and presenting a comprehensive list of results on Hardy and Bergman means. The goal is to provide rapid access to the frontiers of research in this field. This monograph will allow researchers to get acquainted with essentials on inner functions, and it is self-contained, which makes it accessible to graduate students.
This monograph offers an introduction to finite Blaschke products and their connections to complex analysis, linear algebra, operator theory, matrix analysis, and other fields. Old favorites such as the Carathéodory approximation and the Pick interpolation theorems are featured, as are many topics that have never received a modern treatment, such as the Bohr radius and Ritt's theorem on decomposability. Deep connections to hyperbolic geometry are explored, as are the mapping properties, zeros, residues, and critical points of finite Blaschke products. In addition, model spaces, rational functions with real boundary values, spectral mapping properties of the numerical range, and the Darlington synthesis problem from electrical engineering are also covered. Topics are carefully discussed, and numerous examples and illustrations highlight crucial ideas. While thorough explanations allow the reader to appreciate the beauty of the subject, relevant exercises following each chapter improve technical fluency with the material. With much of the material previously scattered throughout mathematical history, this book presents a cohesive, comprehensive and modern exposition accessible to undergraduate students, graduate students, and researchers who have familiarity with complex analysis.