MA6150 Basic Operator theory (modified)
Unit I: Operators on Hilbert spaces: self-adjoint, normal, unitary, isometry, partial isometry, projections, positive operators. (7 Lectures)
Unit I: Spectral Results: Eigen spectrum and spectrum; spectral radius formula; spectral mapping theorem; spectrum of various operators on Hilbert spaces. (12 Lectures)
Unit III: Finite rank operators, compact operators; Riesz-Schauder theory for compact operators; spectral theorem for compact self-adjoint and compact normal operators; singular value decomposition of compact operators; Trace class & Hilbert Schmidt operators. (11 Lectures)
Unit IV: Continuous functional calculus; Polar decomposition; Borel functional calculus; Spectral theorem for self-adjoint & normal operators - sketch of the proofs. (10 Lectures)
1. M. T. Nair, Functional analysis: A First Course, PHI-Learning, New Delhi, 2002 (Fourth Print: 2014).
2. M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. I: Functional Analysis, Academic Press, New York, 1962.
1. J. B. Conway, A Course in Functional Analysis, Springer, 1997.
2. C.D. Kubrusly, Elements of Operator Theory, Birkhauser, 2001.
3. B.D. MacCluer, Elementary Functional Analysis, Springer, 2009.
4. V. S. Sunder, Functional Analysis: Spectral Theory, Hindustan Book Agency (TRIM Series), 1997.