Functional calculus is a mathematical framework used to extend the concept of functions of a real variable to functions of operators. This approach is particularly useful in functional analysis and operator theory. In essence, it allows the application of algebraic operations and functions, such as polynomials or exponentials, to operators on a Hilbert space or Banach space. Functional calculus plays a crucial role in quantum mechanics, spectral theory, and differential equations, where operators often represent physical observables or evolution processes. The most common forms are the polynomial functional calculus and the more advanced holomorphic functional calculus. These tools enable the analysis and manipulation of operator functions, facilitating solutions to complex mathematical and physical problems by extending familiar concepts of functions to an operator setting. This book provides comprehensive insights into the field of functional calculus. It presents the complex subject of functional calculus in the most comprehensible and easy to understand language. Students, researchers, experts and all associated with mathematics will benefit alike from this book. The aim of this book is to present a broad overview of the theory and applications related to functional calculus. The book is based on two main subject areas: matrix calculus and applications of Hilbert spaces. Determinantal representations of the core inverse and its generalizations, new series formulas for matrix exponential series, results on fixed point theory, and chaotic graph operations and their fundamental group are contained under the umbrella of matrix calculus. In addition, numerical analysis of boundary value problems of fractional differential equations are also considered here. In addition, reproducing kernel Hilbert spaces, spectral theory as an application of Hilbert spaces, and an analysis of PM10 fluctuations and optimal control are all contained in the applications of Hilbert spaces. The concept of this book covers topics that will be of interest not only for students but also for researchers and professors in this field of mathematics.
Dan Kucerovsky professor emeritus at the George Washington University specializes in functional analysis, particularly bounded operators on Hilbert spaces. Kucerovsky earned his B.S. from Loyola University and Ph.D. from Louisiana State University under the direction of Heron Collins in 1965, with a dissertation on The Strict Topology and Compactness in the Space of Measures. He has had 20 students who obtained doctorates under his supervision, most of them at Indiana University. He served on the faculty there from 1965 to 1990, when he became head of the mathematics department at the University of Tennessee.
| Preface Chapter 1. Foundations of Functional Analysis Normed Vector Spaces Banach Spaces Inner Product Spaces Hilbert Spaces Linear Operators Spectral Radius Chapter 2. Spectral Theory of Operators Spectrum of an Operator Point, Continuous, and Residual Spectrum Spectral Decomposition Spectral Theorem for Bounded Operators Normal Operators Compact Operators Self-Adjoint Operators Chapter 3. Basics of Functional Calculus Understanding Functional Calculus Motivation and Examples Polynomial Functional Calculus Rational Functional Calculus Continuous Functional Calculus Borel Functional Calculus Taylor Series Chapter 4. Holomorphic Functional Calculus Complex Analysis Review Holomorphic Functions Contour Integration Dunford–Taylor Integral Holomorphic Functional Calculus Cauchy’s Integral Formula for Operators Resolvent Set and Operator-Valued Functions Chapter 5. Spectral Theorem and Functional Calculus Spectral Measures Stone’s Theorem Spectral Integrals Functional Calculus for Self-Adjoint Operators Spectral Mapping Theorem Chapter 6. Borel and Continuous Functional Calculus Borel Measurable Functions Gelfand–Naimark Theorem Commutative C*-Algebras Spectra in C*-Algebras Approximate Identitie |
|
$280.00