Operator Theory

By Prof. G. Ramesh | IIT Hyderabad

Learners enrolled: 316

About the course:
In this course, we aim to study the spectral theory of normal operators and continuous functional calculus. We begin with the introduction of Hilbert space and study bounded operators on these spaces. More often we compare the results on operators with operators on finite-dimensional Hilbert spaces (or matrices). In this way, we study the spectrum and its properties, spectral theorem for compact normal operators which is a generalization of finite-dimensional operators. The further generalization is the spectral theorem for normal operators.

PREREQUISITES : Functional Analysis

INTENDED AUDIENCE : M. Sc Mathematics II Year students and Ph. D first year students

Summary

Course Status :	Completed
Course Type :	Elective
Duration :	12 weeks
Category :	Mathematics
Credit Points :	3
Level :	Postgraduate
Start Date :	25 Jul 2022
End Date :	14 Oct 2022
Enrollment Ends :	08 Aug 2022
Exam Date :	30 Oct 2022 IST

Note: This exam date is subjected to change based on seat availability. You can check final exam date on your hall ticket.

Page Visits

Course layout

Week 1: Review of Hilbert space Theory , Bounded operators on Hilbert spaces, examples

Week 2: Adjoint an operator, examples, Self-adjoint, normal, positive, unitary, isometries, partial isometries
Week 3: Orthogonal projections with examples, invariant subspaces, numerical range and characterization of operators
Week 4: Banach Algebras, inertibility, spectrum
Week 5: Gelfand-Mazur theorem, spectral radius formula, spectral mapping theorem.
Week 6: Subdivion of the spectrum of an operator, properties of the various spectra
Week 7: Computing spectrum with examples
Week 8: Existence of square root, polar-decomposition.
Week 9: Compact operators, properties
Week 10: Spectral theorem for compact self-adjoint operators, spectral theorem for compact normal operators, Schmidt-decomposition, Monotone convergence theorem for self-adjoint operators
Week 11: Spectral theorem for self-adjoint operators, continuous functional calculus, spectral theorem for self-adjoint operators( multiplication form)
Week 12: Spectral theorem for normal operators (both integral and multiplication form), continuous functional calculus for normal operators.

Books and references

1. Conway, John B. A course in functional analysis. Second edition. Graduate Texts in Mathematics, 96. Springer-Verlag, New York, 1990. xvi+399 pp.

2. W. Rudin, {\it Functional analysis}, second edition, International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991.

3. Bhatia, Rajendra Notes on functional analysis. Texts and Readings in Mathematics, 50. Hindustan Book Agency, New Delhi, 2009. x+237 pp. ISBN: 978-81-85931-89-0

4. Gohberg, Israel; Goldberg, Seymour; Kaashoek, Marinus A. Basic classes of linear operators. Birkhäuser Verlag, Basel, 2003. xviii+423 pp. ISBN: 3-7643-6930-2

5. Bachman, George; Narici, Lawrence Functional analysis. Reprint of the 1966 original. Dover Publications, Inc., Mineola, NY, 2000. xii+532 pp. ISBN: 0-486-40251-7

6. Pedersen, Gert K. Analysis now. Graduate Texts in Mathematics, 118. Springer Verlag, New York, 1989. xiv+277 pp.

Instructor bio

Prof. G. Ramesh

IIT Hyderabad

Dr. G. Ramesh completed his Ph. D (Operator Theory) from IIT Madras in 2008 and did Post Doctoral studies at ISI Bangalore, Bengaluru during 2009-2010 and worked as an Assistant Professor at Hyderabad University till June 2011. In 2011 July, he joined as an assistant professor in IUT Hyderabad. Currently, he is an Associate Professor at IIT Hyderabad. His area of specialization is Functional Analysis and Operator Theory.

Course certificate

The course is free to enroll and learn from. But if you want a certificate, you have to register and write the proctored exam conducted by us in person at any of the designated exam centres.

The exam is optional for a fee of Rs 1000/- (Rupees one thousand only).

Date and Time of Exams: 30 October 2022 Morning session 9am to 12 noon; Afternoon Session 2pm to 5pm.

Registration url: Announcements will be made when the registration form is open for registrations.

The online registration form has to be filled and the certification exam fee needs to be paid. More details will be made available when the exam registration form is published. If there are any changes, it will be mentioned then.

Please check the form for more details on the cities where the exams will be held, the conditions you agree to when you fill the form etc.

CRITERIA TO GET A CERTIFICATE

Average assignment score = 25% of average of best 8 assignments out of the total 12 assignments given in the course.

Exam score = 75% of the proctored certification exam score out of 100

Final score = Average assignment score + Exam score

YOU WILL BE ELIGIBLE FOR A CERTIFICATE ONLY IF AVERAGE ASSIGNMENT SCORE >=10/25 AND EXAM SCORE >= 30/75. If one of the 2 criteria is not met, you will not get the certificate even if the Final score >= 40/100.

Certificate will have your name, photograph and the score in the final exam with the breakup.It will have the logos of NPTEL and IIT Hyderabad. It will be e-verifiable at nptel.ac.in/noc.

Only the e-certificate will be made available. Hard copies will not be dispatched.

Once again, thanks for your interest in our online courses and certification. Happy learning.

- NPTEL team