NCMW - Operator Theory and Operator Algebra (2023)

Speakers and Syllabus

 Syllabus to be covered in terms of modules of 6 lectures each:

Name of the Speakers with their affiliation.

No. of Lectures

Detailed Syllabus

B. V. Rajarama Bhat (BVR)

Stat-Math Unit, Indian Statistical Institute, Bangalore 560059, Karnataka

2

 Trace Inequalities
Set-theoretic version of some basic dilation results of operator theory. The lectures would cover Wold decomposition, Halmos dilation, Sz. Nagy dilation,inter-twining lifting, commuting and non-commuting dilations,Completely positive maps, Dilation of Hilbert space operators. There will be also discussion of variations on this theme.

K. B. Sinha (KBS)

J. N. Centre for Advanced Scientific Research, Jakkur, Bangalore560064, India, and Indian Statistical Institute, Karnataka

 

2

Gadadhar Misra (GM)

 

Discipline of Mathematics,

Indian Institute of Technology, Palaj

Gandhinagar 382355, Gujarat

 

2

Jaydeb Sarkar (JS)

Stat-Math Unit, Indian Statistical Institute, Bangalore 560059, Karnataka

3

 Brief history of operator theory. Relationship of operators and functions. Introduction to Hardy space on the open unit disc.Shift operator and basic properties. Reproducing kernel Hilbert spaces. Hardy space as a reproducing kernel Hilbert space.Shift invariant subspaces of the Hardy space. Beurling's view, Inner-outer factorizations, and examples.
Introduction to model spaces. Commutant lifting theorem.
Schur functions and representations of Schur functions. Applications - Nevanlinna-Pick interpolations (past-present-future, if time permits).

Examples of some basic C*-algebras and classifications. Applications to system theory/electrical engineering (if time permits).

Sameer Chavan (SC)

Department of Mathematics and Statistics, Indian Institute of Technology Kanpur Kanpur 208016, Uttar Pradesh

3

Bipul Saurabh (BS)

Discipline of Mathematics,

Indian Institute of Technology, Palaj

Gandhinagar 382355, Gujarat

3

C*-algebra, spectrum, Unitization, the Gelfand-Naimark theorem, continuous Functional Calculus, the Serre-Swan theorem, K-groups, Bott periodicity, Spectral triples, Fredholm operators, Fredholm modules, K-homology groups, Index pairing.

Apoorva Khare (AK)

Department of Mathematics,

Indian Institute of Science,

Bengaluru 560012, India, Karnataka

3

 (a) Definitions, examples of total positivity (TP) and Total nonnegativity (TN)

(b) Whitney’s density theorem: TP is dense in TN

(c) Eigenvalues of square TP/TN matrices are positive/nonnegative.

(d) Variation diminishing property and its history

(e) Polya frequency sequences: examples, generating functions, classification

(f) Polya frequency functions: examples, Laplace transform, classification

(g) If time permits: a brief look at the Laguerre-Polya class and Polya-Schur multipliers

 

Projesh Nath Choudhury (PNC)

Discipline of Mathematics,

Indian Institute of Technology, Palaj

Gandhinagar 382355, Gujarat

3

  References:

  1. W. Arveson, An Invitation to C*-Algebras, Graduate Texts in Mathematics, 2011.
  2. J.B Conway, A Course in Functional Analysis, Springer New York, 2019.
  3. A.  Connes, Noncommutative geometry. Academic Press, Inc., San Diego, CA, 1994.
  4. R.G. Douglas, Banach Algebra Techniques in Operator Theory, Graduate Texts in Mathematics, 1998.
  5. R.G. Douglas and V.I. Paulsen, Hilbert Modules Over Function Algebras, Longman Scientific & Technical, 1989.
  6. S. Karlin, Total positivity. Vol. I, Stanford University Press, Stanford, CA, 1968.
  7. A. Khare, Matrix analysis, Pólya frequency functions, and preservers of (total) positivity, Lecture notes.
  8. M. Martin and  M. Putinar, Lectures on Hyponormal Operators, Birkhäuser, 1989.
  9. R.A. Martinez-Avendano and  P. Rosenthal, An Introduction to Operators on the Hardy-Hilbert Space, Graduate Texts in Mathematics, 2007.
  10. B. Sz Nagy, C. Foias, H. Bercovici and L. Kérchy, Harmonic Analysis of Operators on Hilbert Space, Springer New York, 2010.
  11. A. Pinkus, Totally positive matrices, Cambridge University Press, Cambridge, 2010.
  12. M. Rørdam, F. Larsen and N. Laustsen: An Introduction to K-Theory for C*-Algebras,   Cambridge University Press, 2000.

Time Table

 Tentative time-table, mentioning names of the speakers and tutors with their affiliation:

Day

Date

Lecture
1

(10:00
to
11.00)

Tea

(11.00
to
11.25)

Lecture 2
(11.30
to
12.30)

Lunch

(12.30
to
1.55)

Lecture
3
(2.00
to
3.00)

Tea

(3.00
to
3.25)

Lecture 4

(3.30
to
4.30)

Tea

(4.30
to
4.55)

Discussion Hour

(5.00
to
6:00)

 

 

(name of the speaker)

 

(name of the speaker)

 

(name of the speaker)

 

 

 

 

Mon

 

PNC

 

JS

 

BVR

 

KBS

 

JS+BVR

Tues

 

PNC

 

JS

 

BVR

 

KBS

 

JS+BVR

Wed

 

PNC

 

JS

 

BS

 

GM

 

KBS+PNC

Thu

 

AK

 

SC

 

BS

 

GM

 

SC+BS

Fri

 

AK

 

SC

 

BS

 

GM

 

AK+BS

Sat

 

AK

 

SC

 

BS

 

GM

 

SC+GM

 Full forms for the abbreviations of speakers and tutors:

 PNC: Projesh Nath Choudhury

 AK: Apoorva Khare

 JS: Jaydeb Sarkar

 SC: Sameer Chavan

 BS: Bipul Saurabh

 KBS: K. B. Sinha

 BVR: B. V. Rajarama Bhat

 GM: Gadadhar Misra

 

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