AIS  Function Spaces, Operator Theory and Operator Algebras (2023)
Speakers and Syllabus
Name of the Speaker with affiliation 
No. of Lectures 
Detailed Syllabus 
Sundar Sobers IMSc, Chennai 
6 
Basics of C^*algebras, GelfandNaimark theorem, continuous functional calculus, spectral theorem for normal operators, positive linear functionals, representations and the GNS construction, Universal C^*algebras, examples (including universal C^*algebra generated by an isometry, rotational algebra and Cuntz algebra), and if time permits: inductive limits of C^*algebras and AF algebras. 
Ved Prakash Gupta Jawaharlal Nehru University 
6 
Different topologies on the algebra of bounded linear operators on a Hilbert space (strong, weak and σweak), von Neumann’s double commutant theorem and Kaplansky’s density theorem, normal linear functionals, geometry of projections in a von Neumann algebra and type decomposition of von Neumann algebras, abelian von Neumann algebras, the GNS construction, the TomitaTakesaki theorem (for states), examples of von Neumann algebras (for instance, group von Neumann algebras and group measure space construction), and if time permits: discrete crossed products of von Neumann algebras. 
Prahlad Vaidyanathan IISER Bhopal

6 
Projections and Unitary Elements: Homotopy classes of Unitary elements, Equivalence of Projections; K_0 group of a unital C*algebra: Definition, Functoriality, Examples; The functor K_0: Standard Picture of K_0, Half and Split Exactness, Stability; The K_1 group: Definition, Functoriality, Half and Split Exactness, Stability; Examples: Cuntz algebra, Irrational Rotation Algebra, etc. 
Bata Krishna Das IIT Bombay 
6 
Lecture 1  4 : Introduction of kernel functions and construction of the HKHS from a kernel function, Kernel functions and RKHSs associated with power series (this will include plenty of standard examples), Sums and products of kernels. Multiplier algebras on RKHS. Some selected results on the Hardy Hilbert space. Shift on the Hardy Hilbert space, Wold decomposition of isometries, transfer function realization of schur functions and BCL factorization of the Hardy shift.
Lecture 5  6: Isometric dilations of single contractions (Scheffer's construction and Douglas Model) and consequently von Neumann inequality. Ando dilation using BCL model and refined von Neumann inequality on distinguished varieties. Parrot's examples. Few results on dilations of ntuples of commuting contractions if time permits. 
Jaydeb Sarkar IISc Bangalore 
6 
Lecture 1. Basic introduction to operator and function theory. Introduction to H*p spaces. Radial limits. H*2 space and invariant subspaces, Beurling theorem, inner outer factorizations. Lecture 2. Introduction to model spaces, characteristic functions, and commutant lifting theorem. Lecture 3. Toeplitz operators. Schur functions and representations of Schur functions. Lecture 4. NevanlinnaPick interpolations. The CarathéodoryFejér interpolation. Lecture 5. Schur functions and interpolation on polydisc and ball. Lecture 6. Hankel operators. Applications to system theory/electrical engineering (if time permits). 
Manjunath Krishnapur IISc Bangalore 
6 
Models of random matrices such as GUE (Gaussian unitary ensemble), CUE (circular unitary ensemble) and $\beta$tridiagonal random matrices. Fundamental questions in the subject and methods to tackle them. Free probability and its use in firstorder calculations in random matrix theory. Orthogonal polynomials, determinantal processes and finer analysis of eigenvalues. If time permits, we shall also touch upon nonhermitian matrices and Brown measure. (Any probability facts beyond very basics will be covered.) 
Time Table
TimeTable (with names of speakers and course associates/tutors):
Day 
Date 
Lecture 1 (9.30–11.00) 
Tea (11.05 –11.25) 
Lecture 2 (11.30–1.00) 
Lunch (1.05–2.25) 
Tutorial (2.30–3.30) 
Tea (3.353.55) 
Tutorial (4.005.00) 
Snacks 5.055.30 
Mon 
12 
SS+AG +SP 
BKD+SP+AG 
SS+ AG+SP 
BKD+SP+AG 

Tues 
13 
SS+AG+SP 
BKD+SP+AG 
SS+AG+SP 
BKD+SP+AG 

Wed 
14 
SS+AG+SP 
BKD+SP+AG 
SS+AG+SP 
BKD+SP+AG 

Thu 
15 
SS+AG+SP 
BKD+SP+AG 
SS+AG+SP 
BKD+SP+AG 

Fri 
16 
SS+AG+SP 
BKD+SP+AG 
SS+AG+SP 
BKD+SP+AG 

Sat 
17 
SS+AG+SP 
BKD+SP+AG 
SS+AG+SP 
BKD+SP+AG 

SUNDAY: OFF 

Mon 
19 
PV+SP+AG 
JS+AG+SP 
PV +SP+AG 
JS+AG+SP 

Tues 
20 
PV+SP+AG 
JS +AG+SP 
PV +SP+AG 
JS+AG+SP 

Wed 
21 
PV +SP+AG 
JS +AG+SP 
PV +SP+AG 
JS+AG+SP 

Thu 
22 
PV +SP+AG 
JS +AG+SP 
PV +SP+AG 
JS+AG+SP 

Fri 
23 
PV +SP+AG 
JS +AG+SP 
PV +SP+AG 
JS+AG+SP 

Sat 
24 
PV +SP+AG 
JS +AG+SP 
PV +SP+AG 
JS+AG+SP 

SUNDAY: OFF 

Mon 
26 
VPG+AG+SP 
MK+SB+AG 
VPG+AG+SP 
MK+SB+AG 

Tues 
27 
VPG+AG+SP 
MK+SB+AG 
VPG+AG+SP 
MK+SB+AG 

Wed 
28 
VPG+AG+SP 
MK+SB+AG 
VPG+AG+SP 
MK+SB+AG 

Thu 
29 
VPG+AG+SP 
MK+SB+AG 
VPG+AG+SP 
MK+SB+AG 

Fri 
30 
VPG+AG+SP 
MK+SB+AG 
VPG+AG+SP 
MK+SB+AG 

Sat 
01 
VPG+AG+SP 
MK+SB+AG 
VPG+AG+SP 
MK+SB+AG 
Tutorial Assistants:
S. No. 
Name 
Affiliation 
1 
Mr. Samir Panja 
IIT Bombay 
2 
Dr. Anindya Ghatak 
ISI Bangalore 
3 
Ms. Sudeshna Bhattacharjee 
IISc Bangalore 
Full forms for the abbreviations of speakers and tutors:
JS: Jaydeb Sarkar
BKD: Bata Krishna Das
VPG: Ved Prakash Gupta
MK: Manjunath Krishnapur
SS: Sundar Sobers
PV: Prahlad Vaidyanathan
SP: Samir Panja
AG: Anindya Ghatak
SB: Sudeshna Bhattacharjee