Advanced Training School in Mathematics for Lecturers(ATML)

in Functional Analysis-II

(Sponsored by National Board for Higher Mathematics )

3-15 November, 2008
Resource Persons:
Speakers
Dr. H. Bhate Univ. of Pune hbhate at math.unipune.ernet.in
Dr. Sameer Chavan HRI, Allhabad chavansameer at mri.ernet.in
Dr. Anandateertha Mangasuli  BIM, Pune anandateertha at gmail.com
Dr. V. M. Sholapurkar S.P. College, Pune  deepaksholapurkar at yahoo.com


Unity of Mathematics Lectures:

Prof. G. Misra, I.I.Sc. Bangalore
Bergman Kernels
Dipendra Prasad, TIFR, Bombay
Harmonic Analysis on groups


Course Associate

1. Geetanjali Phatak

2. Pratul Gadagkar

Syllabus

(a) Review of Linear Algebra : Spectral Theorem for Normal Operators on Finite Dimensional Inner Product Spaces.

(b) Hilbert Spaces :
i. Examples of Hilbert Spaces.
ii. Cauchy-Schwarz Inequality, Pythagoras Theorem, Parallelogram
Law, orthogonal complement, orthogonal projection and orthonor- mal basis
iii. Reisz Representation Theorem
iv. Direct Sum of Hilbert Spaces

(c) Operators on Hilbert Spaces :
i. Examples of Operators : Unilateral and bilateral shift, weighted shifts, multiplication operator, integral operator, Fourier transform
ii. Adjoint of an operator, invariant and reducing subspaces
iii. Special classes of operators: finite rank operators, compact operators, isometries, projections, unitary operators, self adjoint operators, normal operators.
iv. Spectrum of an operator : Spectral parts, properties of spectra, computation of spectra of some operators.
v. Properties of spectrum of a compact operator and spectrum of a normal operator
vi. Spectral Theorem for compact normal operators on Hilbert Spaces vii. Applications of Spectral Theorem.

References :
(a) John. B. Conway, A Course in Function Analysis, 2nd edition, (GTM 96), New York, Springer, 1990
(b) Paul R. Halmos, A Hilbert Space Problem Book, 2nd edition, New York, Springer, 1982
(c) Balmohan V. Limaye, Functional Analysis, New Delhi : Wiley Eastern Ltd, 1981