Resource Persons:
Speakers

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

Speakers

[A]A. Mangasuli - Spectral Theorem for Normal Operators on Finite Dimensional Inner Product Spaces

[B]H. Bhate - Hilbert Spaces & Fourier analysis on finite groups
[C]V. Sholapurkar - Bounded operators on Hilbert Spaces
[D]S. Chavan - Spectral Theorem for bounded normal operators on Hilbert Spaces

[UM-1,2,3] G. Misra - Bergman Kernels
[UM-4] Dipendra Prasad - Harmonic Analysis on Groups

Schedule of LecturesFirst Week

Second Week