Operator Theory

About the course:In this course, we aim to study the spectral theory of normal operators and continuous functional calculus. We begin with the introduction of Hilbert space and study bounded operators on these spaces. More often we compare the results on operators with operators on finite-dimensional Hilbert spaces (or matrices). In this way, we study the spectrum and its properties, spectral theorem for compact normal operators which is a generalization of finite-dimensional operators. The further generalization is the spectral theorem for normal operators.PREREQUISITES :Functional AnalysisINTENDED AUDIENCE:M. Sc Mathematics II Year students and Ph. D first year students

Week 1: Review of Hilbert space Theory , Bounded operators on Hilbert spaces, examples Week 2:Adjoint an operator, examples, Self-adjoint, normal, positive, unitary, isometries, partial isometries Week 3:Orthogonal projections with examples, invariant subspaces, numerical range and characterization of operators Week 4:Banach Algebras, inertibility, spectrum Week 5:Gelfand-Mazur theorem, spectral radius formula, spectral mapping theorem. Week 6:Subdivion of the spectrum of an operator, properties of the various spectra Week 7:Computing spectrum with examples Week 8:Existence of square root, polar-decomposition. Week 9:Compact operators, properties Week 10:Spectral theorem for compact self-adjoint operators, spectral theorem for compact normal operators, Schmidt-decomposition, Monotone convergence theorem for self-adjoint operators Week 11:Spectral theorem for self-adjoint operators, continuous functional calculus, spectral theorem for self-adjoint operators( multiplication form) Week 12:Spectral theorem for normal operators (both integral and multiplication form), continuous functional calculus for normal operators.