Spectral Theory and Functional Calculus of Compact Operators

This book addresses the theory of compact operators, a fundamental class of linear operators that occupies a central place in functional analysis and plays a crucial role in establishing key results across various branches of modern mathematics.The work is organized into three main chapters. The first chapter lays the functional analytic foundation by presenting a rigorous treatment of Hilbert spaces, covering their essential properties, as well as the notions of orthogonality and orthonormality. The second chapter is devoted to the study of bounded linear operators on Hilbert spaces, examining their fundamental theorems and properties, with particular attention to adjoint operators, including their definitions and characterizations. The third chapter constitutes the core of this book, providing a comprehensive and self-contained exposition of compact operators, their structural properties, spectral theory, functional calculus, and the most significant theorems associated with them, including the Fredholm theorem and the Stone-Weierstrass theorem.