MA6605: Operator Theory

Course Details:

Operators on Hilbert spaces: self-adjoint, normal, unitary, isometry, partial isometry, projections, positive operators

Spectrum of operators: spectral radius formula, spectral mapping theorem

Compact operators: Spectrum of compact operators; spectral theorem for compact self-adjoint and compact normal operators; singular value decomposition of compact operators; Trace class & Hilbert Schmidt Operators

Statement of Spectral Theorem for self adjoint and normal operators, Polar decomposition, Continuous functional calculus

Text Books:

1. J. B. Conway, A course in functional analysis, GTM (96), Springer, 2007
2. M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. I: Functional Analysis, Academic Press, New York, 1981
3. B. V. Limaye, Functional Analysis, 3rd Ed., New Age International Publishers, 2014

Reference Books:

1. C.D. Kubrusly, Elements of Operator Theory, Birkhauser, 2001
2. V. S. Sunder, Operators on Hilbert Space, Springer, 2016
3. W. Rudin, Functional Analysis, 2nd edn, McGraw Hill Education, 2017
4. C. W. Groetsch: Elements of applicable functional analysis, 2nd edition, M. Dekker, 1980
5. Rajendra Bhatia, Notes on Functional Analysis, Hindustan Book Agency, 2015
6. N. Dunford and J. T. Schwartz, Linear operators- I: General theory, Wiley-Interscience, New York, 1988
7. N. Dunford and J. T. Schwartz, Linear operators- II: Spectral Theory, Self Adjoint Operators in Hilbert Space, Wiley-Interscience, New York, 1988
8. N. Dunford and J. T. Schwartz, Linear operators- III: Spectral Operators, Wiley-Interscience, New York, 1988
9. B. MacCluer, Elementary Functional Analysis, Springer-Verlag New York, 2009
10. K. Yoshida, Functional Analysis, 6th edn, Springer, 1995