AIS - Operator Theory (2019)

Speakers and Syllabus


Name of the Speaker with affiliation No. of Lectur es Detailed Syllabus
K. B. Sinha (KBS)
J. N. Centre for Advanced Scientific Research, Jakkur, Bangalore560064, India, and Indian Statistical Institute, India
4 Unbounded Operators in Hilbert Space and Applications in Quantum Mechanics: Basic results/examples of unbounded, closed and selfadjoint operators, Sum of two unbounded operators, some results/examples, the spectral theory of such operators, Spectral theorem of Unbounded self-adjoint operators, examples with computation of spectral measure/families, the types of Spectrum, Computation of spectra for differential operators, Dirichlet vs Neumann Operators, Sum of two unbounded self-adjoint operators (if time permits).
B. V. Rajarama Bhat (BVR)
Stat-Math Unit, Indian Statistical Institute,
Bangalore 560059
4 Completely positive maps and dilation theorems: Positive maps, Examples and basic properties, von Neumann inequality, Fejer-Riesz Lemma and its application to trigonometric moments, Completely Positive maps, Schur product and tensor product, Choi Theorem and its application to numerical range, Stinespring’s Dilation Theorem and its applications (if time permits).
Partha Sarathi Chakraborty (PC)
Stat-Math Unit, Indian Statistical Institute, Kolkatta 700 108
4 Introduction to compact Quantum groups: Definition of compact quantum groups and its examples, the Haar state, right regular representations, irreducible unitary representations, the Hopf*- algebra of matrix elements, dual of compact quantum groups, glimpse into locally compact quantum groups.
Shibanand Biswas (SB)
Indian Institute of Science Education And Research Kolkata 741 246
4 Operator theory approach to Pick Interpolation problem: Kernels and function spaces, Hardy spaces, Fatou's theorem, Examples of spaces with H-infinity as multiplier algebra, Necessity of positivity of Pick matrix, Szego kernel having Pick property and its uniqueness, realization formula and Pick problem, Characteriztion of kernels with complete pick property and examples.
Satyajit Guin (SG)
Department of Mathematics and Statistics Indian Institute of Technology Kanpur Kanpur 208016 Uttar Pradesh, India
4 Representation theory for compact groups: Compact and locally compact groups, the Haar measure, Fourier transform, C*-algebra, spectrum, Unitization, the Gelfand-Naimark theorem, continuous Functional Calculus, The dual group, C*-algebra of an LCA group, Pontryagin duality, representations of compact groups.
Sameer Chavan (SC)
Department of Mathematics and Statistics Indian Institute of Technology Kanpur Kanpur 208016 Uttar Pradesh, India
4 Introduction to graph-theoretic operator theory: Directed graphs/trees, examples and basic properties, weighted shifts on directed trees, motivation, examples and basic properties (boundedness, adjoint, circularity, kernel and cokernel), reproducing kernel Hilbert spaces of holomorphic functions associated with directed trees, bounded point evaluation, spectral theory (if time permits).

 References*:

  1. 1. Barry Simon, Operator theory. A Comprehensive Course in Analysis, Part 4. American Mathematical Society, Providence, RI, 2015. xviii+749 pp.
  2. Vern Paulsen, Completely bounded maps and operator algebras. Cambridge Studies in Advanced Mathematics, 78. Cambridge University Press, Cambridge, 2002. xii+300 pp.
  3. Jim Agler, J. E. McCarthy, Pick interpolation and Hilbert function spaces. (English summary) Graduate Studies in Mathematics, 44. American Mathematical Society, Providence, RI, 2002. xx+308 pp.
  4. Z. Jablonski, Il Bong Jung, Jan Stochel, Weighted shifts on directed trees. Mem. Amer. Math. Soc. 216 (2012), no. 1017, viii+106 pp.
  5. S. L. Woronowicz, Compact quantum groups. Symétries quantiques (Les Houches, 1995), 845–884, North-Holland, Amsterdam, 1998.
  6. Folland, Gerald B. A course in abstract harmonic analysis. Second edition. Textbooks in Mathematics. CRC Press, Boca Raton, FL, 2016. xiii+305 pp

*Lecture notes from the speakers will be distributed.

Tutorial Assistants:

Sr.n Name Affiliation
1 Rajeev Gupta (RG) IIT Kanpur
2 Shailesh Trivedi (ST) IIT Kanpur
3 Ankit Bhojak (AB) IIT Kanpur

 


Time Table

Time-Table (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
(3.35-3.55)
Tutorial
(4.00-5.00)
Snacks
5.05-5.30
    (name of the speaker )   (name of the speaker)   (name of the speaker + tutors)   (name of the speaker + tutors)  
Mon 16/12/19 SB   SC   SB+RG   SC+ST  
Tues 17/12/19 SB   SC   SB+RG   SC+ST  
Wed 18/12/19 SB   SC   SB+RG   SC+AB  
Thu 19/12/19 SB   SC   SB+RG   SC+AB  
Fri 20/12/19 PC   SG   PC+ST   SG+AB  
Sat 21/12/19 PC   SG   PC+ST   SG+AB  
   SUNDAY : OFF    
Mon 23/12/19 PC   SG   PC+RG   SG+ST  
Tues 24/12/19 PC   SG   PC+RG   SG+ST  
Wed 25/12/19 KBS   BVR   KBS+RG   BVR+AB  
Thu 26/12/19 KBS   BVR   KBS+RG   BVR+AB  
Fri 27/12/19 KBS   BVR   KBS+ST   BVR+AB  
Sat 28/12/19 KBS   BVR   KBS+ST   BVR+AB  
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