The book "The Functional Calculus for Sectorial Operators" by Markus Haase provides a comprehensive treatment of functional calculus, focusing on operators on Banach spaces. It introduces the concept of functional calculus as a method to insert operators into functions, enabling expressions like $ A^{\alpha} $, $ e^{-tA} $, and $ \log A $, where $ A $ is a sectorial operator. The text explores the theory of sectorial operators, their properties, and the associated functional calculus, including the natural functional calculus, the composition rule, and the spectral mapping theorem. It also discusses fractional powers, holomorphic semigroups, and the boundedness of the $ H^{\infty} $-calculus. The book covers topics such as interpolation spaces, the relationship between operators and their functional calculus properties, and applications in various areas like evolution equations and numerical analysis. It includes detailed discussions on Hilbert spaces, similarity theorems, and the connection between functional calculus and maximal regularity. The text is structured into chapters that provide a thorough exploration of the subject, with an appendix offering additional material on operators, interpolation spaces, forms, and the spectral theorem. The book aims to provide a self-contained and comprehensive resource for understanding functional calculus, with a focus on its theoretical foundations and practical applications.The book "The Functional Calculus for Sectorial Operators" by Markus Haase provides a comprehensive treatment of functional calculus, focusing on operators on Banach spaces. It introduces the concept of functional calculus as a method to insert operators into functions, enabling expressions like $ A^{\alpha} $, $ e^{-tA} $, and $ \log A $, where $ A $ is a sectorial operator. The text explores the theory of sectorial operators, their properties, and the associated functional calculus, including the natural functional calculus, the composition rule, and the spectral mapping theorem. It also discusses fractional powers, holomorphic semigroups, and the boundedness of the $ H^{\infty} $-calculus. The book covers topics such as interpolation spaces, the relationship between operators and their functional calculus properties, and applications in various areas like evolution equations and numerical analysis. It includes detailed discussions on Hilbert spaces, similarity theorems, and the connection between functional calculus and maximal regularity. The text is structured into chapters that provide a thorough exploration of the subject, with an appendix offering additional material on operators, interpolation spaces, forms, and the spectral theorem. The book aims to provide a self-contained and comprehensive resource for understanding functional calculus, with a focus on its theoretical foundations and practical applications.