Spectral Theory of Normal Operators

Abstract

We discuss the spectral theory of bounded normal operators on Hilbert Space and functional cal- culus, as well as the Gelfand-Neimark-Segal construction of C ⇤ -algebras, also discuss symmetric extensions of unbounded operators. We begin by introducing the spectral theory for compact self-adjoint operators and then extend it to compact normal operators. We also discuss the idea of the spectrum for Banach algebras and explores complex analysis for operator-valued functions, including integration and Cauchy integral formula. Finally, we discuss the concept of unbounded operators and provides the idea of symmetric self-adjoint extensions of closed symmetric unbounded operators