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.
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