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Ma 111a:  Topics in Analysis (Fall 2016-17)

ANNOUNCEMENTS


COURSE DESCRIPTION

The spectral theory of linear operators plays an important role in Analysis. Using the theory of Banach algebras we will prove the spectral theorem that provides conditions under which a bounded operator can be diagonalized. We will subsequently generalize the results to unbounded operators and discuss some applications. In the last part of the course the theory of semi-groups of operators will be introduced.

Among the topics covered in this course are: Banach algebras, basic properties of spectra, the spectral theorem, unbounded operators, the Cayley transform, semi-groups of operators, the Hille—Yosida theorem.


PREREQUISITES

Basic knowledge of Functional Analysis and Operator Theory


SCHEDULE

TR, 10:30 - 11:55, 351 Sloan


INSTRUCTORS

Lukas Schimmer
HARRY BATEMAN INSTRUCTOR IN MATHEMATICS
380 Sloan
626-395-2891


TA's

n/a


OFFICE HOURS

Thursday 6:00-7:00pm or by appointment per email


POLICIES

Grades

pass/fail

Homework Policy

The examination will take place in the form of student talks on related subjects.


TOPICS COVERED


TEXTBOOKS

[1] Rudin, W.: Functional Analysis, ISBN-13: 978-0070542365

Additional resources:

[2] Simon, B.: Operator Theory: A Comprehensive Course in Analysis, Part 4, ISBN-13: 978-1470411039

[3] Akhiezer, N. I. and Glazman, M.: Theory of Linear Operators in Hilbert Space, ISBN-13: 978-0486677484


LECTURE NOTES

Date Description Reference
09/27/16 Banach spaces, Bounded operators, Banach Algebras [1] pp. 245-252
09/29/16 Complex Homomorphisms [1] pp. 249-253
10/04/16 Basic properties of spectra, Gelafnd--Mazur theorem, Symbolic Calculus [1] pp. 253-262
10/06/16 Spectral mapping theorem, Roots and logarithm,
Ideals and homomorphisms of commutative Banach Algebras
[1] pp. 262-277
10/11/16 Gelfand transforms, B* Algebras [1] pp. 277-286
10/13/16 Gelfand--Naimark Theorem [1] pp. 287-290
10/18/16 Hermitian square roots, Applications to non-commutative algebras [1] pp. 290-296
10/20/16 Resolutions of the identity, Spectral measure [1] pp. 316-321
10/25/16 The spectral theroem for bounded normal operators,
Theorem of Fuglede--Putnam--Rosenblum
[1] pp. 325-321
[1] pp. 315-316
10/27/16 Applications of the spetral theorem for bounded operators:
Characterisation of Fourier--Stiltjes coefficients,
Ergodic systems,
Bohr's theorem for almost periodic functions

[2] pp. 633-634
[1] pp. 339-340
[3] pp. 132-138
11/01/16 Bohr's theorem for almost periodic functions,
The spectrum of normal operators
[3] pp. 132-138
[1] pp. 326-329
11/03/16 Spectrum of compact operators, Square roots of positive operators,
Measure/multiplication operator version of the spectral theorem
[1] pp. 329-333
[2] pp. 293-295
11/08/16 Measure/multiplication operator version of the spectral theorem
Unbounded operators
[2] pp. 294-295
[1] pp. 347-349
11/10/16 Momentum operator, Symmetric operators, Self-adjoint operators [1] pp. 349-354
11/15/16 Maximally symmetric operators, Cayley transform [1] pp. 354-358
11/17/16 Cayley transform, Deficiency Indices, Self-adjoint extensions [1] pp. 358-361
11/22/16 Resolution of the identity, Funcitonal calculus for measurable
functions, Resolvent set, Point/Continuous/Residual sepctrum
[1] pp. 361-367
11/24/16 Thanksgiving  
11/29/16 Spectral Theorem for (unbounded) self-adjopint operators
Dirichlet Laplacian, Neumann Laplacian,
Berezin--Li--Yau inequality,
Semi-groups
[1] pp. 368-369
A. Laptev, J. Funct. Anal. (1997)
[1] p. 375
12/01/16 Hille--Yosida theorem [1] pp. 376-382
12/06/16 Student talks  
12/08/16 End of term  

HOMEWORK

n/a

EXAMS

The examination will take place in the form of student talks on related subjects.


READING

n/a