Analysis Seminar
Speaker: Nilabha Saha, IIT Bombay
Host: Sutanu Roy
Title: Unfolding the Spectrum of Bounded Normal Operators
Time, day and date: 5:00:00 PM - 6:00:00 PM, Monday, November 10
Venue: Ramanujan Hall
Abstract: The spectral theorem provides the bridge between abstract $C^*$-algebra theory and the concrete analysis of operators on Hilbert spaces. Rather than beginning with special cases such as compact or self-adjoint operators, this talk explores the general formulation for normal bounded operators through the structure of commutative $C^*$-algebras.

This talk will develop the spectral theorem in three interrelated formulations. First, we will examine the correspondence between elements of a commutative $C^*$-subalgebra of bounded operators and integrals of their Gelfand transforms against a unique projection valued measure. Second, we will present the representation of such algebras as multiplication operators on suitable $L^2$-spaces, highlighting the role of the measure space constructed from the spectrum. Finally, we will specialise to normal bounded operators and describe a result that assigns to each bounded function a corresponding operator.