Algebraic Structures and Operator Calculus 1st Edition by Feinsilver, René Schott ISBN 9401047200 9789401047203

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Product details:
ISBN 10: 9401047200
ISBN 13: 9789401047203
Author: Feinsilver, René Schott

Algebraic Structures and Operator Calculus 1st Edition Table of contents:

Part I: Basic Algebraic Structures

2. Groups and Group Representations

   • Definition and Examples of Groups
   • Group Homomorphisms and Isomorphisms
   • Group Actions and Representations
   • Applications of Groups in Symmetry Analysis

3. Rings and Modules

   • Basic Definitions of Rings and Fields
   • Properties of Commutative and Non-Commutative Rings
   • Modules and Their Role in Abstract Algebra
   • Quotient Rings and Ideals

4. Vector Spaces and Linear Algebra

   • Properties of Vector Spaces and Subspaces
   • Linear Transformations and Matrices
   • Eigenvalues and Eigenvectors
   • Inner Product Spaces and Orthogonality

5. Algebras and Their Representations

   • Definition and Examples of Algebras
   • Structure of Lie Algebras and Matrix Algebras
   • Representations of Algebras
   • Applications in Quantum Mechanics

Part II: Operator Theory

6. Basic Concepts of Operator Theory

   • Definition of Linear Operators
   • Bounded and Unbounded Operators
   • Operator Norms and Continuity
   • Spectral Theory of Operators

7. Hilbert Spaces and Operators

   • Properties of Hilbert Spaces
   • Bounded Linear Operators on Hilbert Spaces
   • The Spectral Theorem for Self-Adjoint Operators
   • The Role of Operators in Quantum Mechanics

8. Differential Operators and Their Calculus

   • Classical Differential Operators
   • Operator Calculus for Differential Equations
   • Applications to Partial Differential Equations
   • Distribution Theory and Generalized Functions

9. Integral Operators and Their Calculus

   • Basic Properties of Integral Operators
   • Fredholm and Volterra Integral Equations
   • Applications in Integral Equations and Quantum Mechanics

Part III: Algebraic Structures in Operator Calculus

10. Lie Groups and Lie Algebras

    • Definition and Structure of Lie Groups
    • Lie Algebras and their Representations
    • Applications of Lie Groups in Quantum Mechanics
    • The Relationship Between Lie Groups and Operator Theory

11. Commutative and Non-Commutative Operator Algebras

    • Structure of Commutative Algebras
    • Non-Commutative Operator Algebras
    • Von Neumann Algebras and C*-Algebras
    • Applications in Statistical Mechanics

12. Symmetry and Group Theory in Operator Calculus

    • Symmetries in Mathematical and Physical Systems
    • The Role of Symmetry in Quantum Field Theory
    • Group Representations in Quantum Computing
    • Applications of Symmetry in Operator Theory

Part IV: Advanced Topics

13. Spectral Theory and Spectral Decompositions

    • The Spectral Theorem for Bounded Operators
    • Spectral Decompositions of Operators
    • Eigenvalues, Eigenvectors, and the Spectral Radius
    • Applications to Quantum Systems and Signal Processing

14. Operator Algebras in Mathematical Physics

    • The Role of Operator Algebras in Quantum Mechanics
    • Quantum Field Theory and Operators
    • Operator Calculus in Statistical Mechanics
    • Operator Algebras in String Theory

15. Applications to Partial Differential Equations

    • Operator Methods in Solving PDEs
    • The Role of Operators in Boundary Value Problems
    • Functional Analysis and PDEs
    • Applications to Heat, Wave, and Schrödinger Equations

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