**[Cr:4, Lc:3, Tt:1, Lb:0]**

- Recapitulation: Bounded linear operator and their spectrum. Compact operator and their spectral theory, Example of compact operators, Fredholm alternative.
- Commutative Banach algebras: spectrum, resolvent, maximal ideals and Gelfand Theorem.
- Gelfand theory for commutative algebras.
- Resolutions of identity operator and integration with respect to projection valued measure.
- Decomposition of spectrum: absolutely continuous, singular and point spectra.
- Functional calculus for bounded self adjoint operator.
- Spectral resolution of normal and unitary operators.
- Square root of positive operators and polar decomposition of bounded operators.

Additional Topics:

- Introduction to theory of unbounded Operators.

- Walter Rudin, Functional analysis Second edition. International Series in Pure and Applied Mathematics. McGraw-Hill, Inc. (1991).
- Peter D. Lax Functional Analysis, Wiley Interscience (2002).
- K. R. Davidson, -Algebras by Example Text and Readings in mathematics 11, Hindustan Book Agency (1996).
- M. Reed and B. Simon, Methods of Modern mathematical physics I, Functional Analysis, Academic press (1975).
- W. Arveson, An invitation to -algebras, Graduate Texts in Mathematics, No. 39. Springer-Verlag (1976).