Module 1( 8 hours) 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.
Module 2 (8 hours) Orthogonal projections, isometric and unitary operators, finite rank and compact operators, Hilbert-Schmidt operators, self adjoint and normal operators
Module 3 (8 hours) 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.
Module 4 (10 hours) 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.

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.

CO1: Students will learn the Hilbert space and the Riesz representation theorem.
CO2: They will learn the basics of bounded and unbounded operators.
CO3: They will understand the spectral theorem for compact, self-adjoint, and normal operators.
CO4: They will learn the Banach algebra and its properties.
CO5: They will gain the knowledge of C* algebras and the famous Gelfand-Mazur theorem.

R. G. Douglas, Banach Algebra Techniques in Operator Theory, Springer

Carlos S. Kubrusly, Elements of Operator Theory, Birkhäuser

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