Subsections
[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).
