Operator Semigroups

A comprehensive and rigorous guide to the modern theory of operator semigroups and its powerful applications to evolution equations.

Operator Semigroups: Theory and Applications to Evolution Equations provides a complete and systematic journey into one of the most essential tools of modern applied analysis. This text is designed for graduate students, PhD candidates, and researchers in applied mathematics, mathematical physics, and engineering who need to understand and solve time-dependent differential equations.

Beginning with the foundations of functional analysis and operator theory, this book moves systematically from uniformly continuous semigroups to the core of the subject: strongly continuous (C0) semigroups. It provides clear, detailed proofs of the fundamental theorems that govern their behavior.

Key topics covered include:

• The Hille-Yosida Theorem: The cornerstone theorem for characterizing generators of C0-semigroups.

• The Lumer-Phillips Theorem: A vital tool for understanding dissipative operators and contraction semigroups.

• Analytic and Differentiable Semigroups: In-depth treatment of semigroups with enhanced regularity properties, crucial for parabolic equations.

• Approximation and Convergence: A study of the Trotter-Kato theorems and their applications.

• Applications to PDEs: Concrete examples showing how to model and solve classical evolution equations, such as the heat equation, the wave equation, and the Schrödinger equation.

• Non-Autonomous Problems: An advanced look at time-dependent evolution equations, moving beyond the classical autonomous framework.

This book bridges the gap between abstract theory and practical application, making it an indispensable resource for anyone working with partial differential equations, control theory, or mathematical modeling.

• "Ideal for qualifying exams preparation"
• "Self-contained with complete proofs"
• "Prerequisites: Basic functional analysis"
• Mentionnez les applications concrètes (ingénierie, physique quantique, finance)

This is the essential reference for mastering the analysis of infinite-dimensional dynamical systems.