This course is designed to give students an idea of the modern operator theory and its applications. After a short review of some classes of bounded linear operators on a Hilbert space, we will consider projection operators and the spectral family. Using the spectral family, spectral representation of self adjoint operators will be obtained. Then we will turn to the theory of unbounded linear operators. Spectral representations of unitary and consequently, not necessarily bounded self adjoint operators will be discussed. Finally, we will consider applications of unbounded operators in Quantum Mechanics and in particular the Heisenberg uncertainty principle.
The course is aimed at Mathematics students who want to pursue a career in Analysis and its applications.