Course Details
Subject {L-T-P / C} : MA5116 : Operator Theory { 3-0-0 / 3}
Subject Nature : Theory
Coordinator : Dr. Sangita Jha
Syllabus
Introduction to Hilbert space, The Riesz-Representation Theorem, The existence of orthogonal bases, The dimension of Hilbert spaces,Bounded operators on Hilbert spaces, adjoints of bounded operators, algebra of bounded operators Orthogonal projections, isometric and unitary operators, finite rank and compact operators, Hilbert-Schmidt operators, self adjoint and normal operators Spectra of bounded operators, invariant and reducing subspaces Spectral theorem for compact operators, polar and singular value decompositions, Spectral theorem for bounded self adjoint and normal operators. Banach algebras: The Banach algebra of continuous functions, Abstract of Banach algebras, Abstract index in a Banach algebra, The space of multiplicative linear functions, The Gelfand transform, The Gelfand Mazur theorem, The Gelfand theorem for commutative Banach algebras, The bilateral shift operators, C*-algebras.
Course Objectives
- Introduction to Operator theory
- To study operators in Hilbert spaces and understand the Spectral Theorem
- To provide the connection in other fields like quantum mechanics.
Course Outcomes
Learn some introductory concepts about operator theory. After completing the course students will understand the Spectral theorem.
Essential Reading
- R. G. Douglas, Banach Algebra Techniques in Operator Theory, Springer
- Carlos S. Kubrusly, Elements of Operator Theory, Birkhäuser
Supplementary Reading
- Y. A. Abramovich, C. C. Aliprantis, An Invitation to Operator Theory, American Mathematical Society
- John B. Conway, A Course in Operator Theory, American Mathematical Society