The Spectral Theorem stands as one of the central pillars of functional analysis, unifying linear algebra, operator theory, and measure theory. This mini-course offers a systematic journey through the spectral theory of normal operators, tracing its development from finite-dimensional linear algebra to the infinite-dimensional Hilbert space setting.
We will begin by reviewing the finite-dimensional spectral theorem for normal matrices, building intuition for the infinite-dimensional generalizations that follow. From there, we will explore compact normal operators before treating general bounded normal operators.
Finally, we discuss the three classical, equivalent formulations of the Spectral Theorem:
•The Continuous Functional Calculus version
•The Multiplication Operator representation
•The Spectral Measure (Projection-Valued Measure) formulation
Time permitting, we will also highlight key historical developments.
The series is designed to be accessible to graduate students with a basic background in Hilbert space theory and measure theory.
Dr. K. Sumesh is a faculty member in the Department of Mathematics at IITM.
