Contents



Preface

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

Functional analysis is the most important branch of mathematics, whose foundation was laid by the great Persian polymath Muhammad ibn Mūsā al-Khwārizmī, also known as Algorithmi, during 973–1048. He named this branch the "Theory of Functions." Later, Newton and Leibnitz enriched this branch by introducing the concept of derivatives and integrals during 1665–1742 and thus gave birth to another name: calculus. This branch of mathematics has been recently divided into several subbranches, including differential calculus, integral calculus, stochastic calculus, etc. In mathematics, a functional calculus is a theory that permits someone to apply mathematical functions to mathematical operators. Now, functional calculus is a branch that connects operator theory, classical calculus, algebra, and functional analysis. In daily life, functionals are increasingly used to model real-world situations, for example if *f: R→R* is real valued functional from real to real number system. If we apply *f* on some function *x*∈*R*, then *f(x)* makes no sense but if we write it in equation form, then it makes sense, e.g. *f(x)= x*, which represents a physical process between two quantities such that there is direct proportionality. Similar problems occur daily in our surroundings. Therefore, it is necessary to understand what criteria should be satisfied by concerned functionals and operators used in modeling or in the description of daily life problems. It is functional calculus that guides and provides us with the path to how, when, and where particular functionals and operators may be used. Mostly, integral and differential equations are used when we wish to solve a technique or procedure that converts the mentioned equations into algebraic equations of known and unknown functions and functionals. Keeping these needs in mind, the editor of this book has been motivated to welcome international mathematicians and researchers to contribute various topics that address the areas of functional calculus and its applications in both pure and applied analysis. The editor has incorporated contributions from a diverse group of leading

researchers in the field of functional calculus. This book aims to provide an overview of the present knowledge that addresses applications and results related to functional calculus. The main topics covered in this book are determinantal representations of the core inverse and its generalizations, which provides a foundation to solve matrix equations. Furthermore, new series formulae for matrix exponential series have been developed, which are used in solving algebraic equations. Also covered are results on fixed point theory, which is used for mapping the satisfying condition (DA) in Banach space. Results that address folding on chaotic graph operations and their fundamental groups are also introduced. Such algebraic structures are largely used in biology and chemistry. Elsewhere in the book, a brief review is considered of Hilbert space with its fundamental features and features of reproducing kernels in corresponding spaces. Spectral theory is an important area that is most applicable in quantum mechanics. Therefore, a number of fundamental concepts have been investigated regarding analytical applications and observations of PM10 fluctuations. Optimal control is a very important procedure, which is increasingly used in the study of mathematical models of real-world problems. It is helpful in developing future

calculus and applications of Hilbert spaces.

*by Kamal Shah,Thabet Abdeljawad, Hammad Khalil and Rahmat Ali Khan*
