Preface

It gives me great pleasure to introduce as well as discuss, learn, solve, teach, and work with operator theory. This book discusses some fundamental aspects of the theory of operators and explores their use to solve a large array of problems. The book addresses many topics, including the paradigm of complex probability, principal parts extension for a Noether operator A, the sets of fractional operators and some of their applications, the total and partial differentials as algebraically manipulable entities, and more.

In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operators or closed operators; consideration may also be given to nonlinear operators. The study of operator theory, which depends heavily on the topology of function spaces, is a branch of functional analysis. Also, if a collection of operators forms an algebra over a field, then it is an operator algebra. Hence, the description of operator algebras is part of operator theory.

Moreover, single-operator theory deals with the properties and classification of operators, considered one at a time. For example, the classification of normal operators in terms of their spectra falls into this category. In addition, the theory of operator algebras brings algebras of operators such as C\*-algebras to the fore.

Many operators that are studied are operators on Hilbert spaces of holomorphic functions, and the study of the operator is intimately linked to questions in function theory. For example, Beurling's theorem describes the invariant subspaces of the unilateral shift in terms of inner functions, which are bounded holomorphic functions on the unit disk with unimodular boundary values almost everywhere on the circle. Beurling interpreted the unilateral shift as multiplication by the independent variable on the Hardy space. The success in studying multiplication operators, and more generally Toeplitz operators (which are multiplication, followed by projection onto the Hardy space), has inspired the study of similar questions in other spaces, such as the Bergman space. Therefore, operator theory has a connection with complex analysis.

This volume illustrates the use of operator theory when applied to solve specific problems and discusses some fundamental aspects of pure and applied operator theory. It is a useful resource for scholars, researchers, and undergraduate and graduate students in pure and applied mathematics, classical and modern physics, engineering, and science in general.

**Abdo Abou Jaoudé, Ph.D.**

Associate Professor, Mathematics and Statistics, Notre Dame University-Louaizé, Zouk Mosbeh, Lebanon

**Chapter 1**

of MCPP

*Abdo Abou Jaoudé*

*philosophy."*

**Abstract**

**1**

*"Subtle is the Lord. Malicious, He is not."*

*"Ex nihilo nihil fit*: *Nothing comes from nothing.*"

*"All is One. From One all things.*"

The Paradigm of Complex

Probability and the Theory of

Metarelativity: A Simplified Model

*"Mathematics, rightly viewed, possesses not only truth but supreme beauty … "*

*"There are more things in heaven and earth, Horatio, than are dreamt of in your*

All our work in classical probability theory is to compute probabilities. The original idea in this research work is to add new dimensions to our random experiment, which will make the work deterministic. In fact, probability theory is a nondeterministic theory by nature; which means that the outcome of the events is due to chance and luck. By adding new dimensions to the event in the real set of probabilities **R**, we make the work deterministic, and hence a random experiment will have a certain outcome in the complex set of probabilities and total universe **G = C**. It is of great

*"Logic will get you from A to Z; imagination will get you everywhere."*

*Albert Einstein.*

*Bertrand Russell.*

*Albert Einstein.*

*Parmenides.*

*Heraclitus.*

*Hamlet (1601), William Shakespeare.*

**Chapter 1**
