The book "Operator Theory: Advances and Applications" (Vol. 169) is authored by Markus Haase and published by Birkhäuser Verlag. The main focus of the book is functional calculus, which involves inserting operators into functions to define expressions like \( A^\alpha \), \( e^{-tA} \), and \( \log A \), where \( A \) is an unbounded operator on a Banach space. The author emphasizes the importance of understanding the form of knowledge rather than just stating and proving theorems, reflecting a deep philosophical stance. The book is structured into several chapters, each covering different aspects of functional calculus: 1. **Axiomatics for Functional Calculi**: Introduces the concept of functional calculus and provides an abstract framework. 2. **The Functional Calculus for Sectorial Operators**: Discusses sectorial operators and their functional calculus, including the natural functional calculus and the composition rule. 3. **Fractional Powers and Semigroups**: Explores fractional powers and holomorphic semigroups. 4. **Strip-type Operators and the Logarithm**: Focuses on strip-type operators and the logarithm of operators. 5. **The Boundedness of the \( H^\infty \)-calculus**: Treats the boundedness of the \( H^\infty \)-calculus and perturbation theory. 6. **Interpolation Spaces**: Investigates real and complex interpolation spaces and their relation to functional calculus. 7. **The Functional Calculus on Hilbert Spaces**: Examines numerical range conditions and similarity problems in Hilbert spaces. 8. **Differential Operators**: Studies elliptic operators and their relation to Fourier multiplier theory. 9. **Mixed Topics**: Includes applications to time-discretization schemes of parabolic equations and the maximal regularity problem. The book also includes appendices covering operator theory, interpolation spaces, Hilbert spaces, and approximation theory. The author acknowledges the support of various institutions and individuals, including colleagues, friends, and family, and dedicates the book to his father.The book "Operator Theory: Advances and Applications" (Vol. 169) is authored by Markus Haase and published by Birkhäuser Verlag. The main focus of the book is functional calculus, which involves inserting operators into functions to define expressions like \( A^\alpha \), \( e^{-tA} \), and \( \log A \), where \( A \) is an unbounded operator on a Banach space. The author emphasizes the importance of understanding the form of knowledge rather than just stating and proving theorems, reflecting a deep philosophical stance. The book is structured into several chapters, each covering different aspects of functional calculus: 1. **Axiomatics for Functional Calculi**: Introduces the concept of functional calculus and provides an abstract framework. 2. **The Functional Calculus for Sectorial Operators**: Discusses sectorial operators and their functional calculus, including the natural functional calculus and the composition rule. 3. **Fractional Powers and Semigroups**: Explores fractional powers and holomorphic semigroups. 4. **Strip-type Operators and the Logarithm**: Focuses on strip-type operators and the logarithm of operators. 5. **The Boundedness of the \( H^\infty \)-calculus**: Treats the boundedness of the \( H^\infty \)-calculus and perturbation theory. 6. **Interpolation Spaces**: Investigates real and complex interpolation spaces and their relation to functional calculus. 7. **The Functional Calculus on Hilbert Spaces**: Examines numerical range conditions and similarity problems in Hilbert spaces. 8. **Differential Operators**: Studies elliptic operators and their relation to Fourier multiplier theory. 9. **Mixed Topics**: Includes applications to time-discretization schemes of parabolic equations and the maximal regularity problem. The book also includes appendices covering operator theory, interpolation spaces, Hilbert spaces, and approximation theory. The author acknowledges the support of various institutions and individuals, including colleagues, friends, and family, and dedicates the book to his father.