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Spectral theory Description

Key Elements

Code

MATH 417

Formation

M1 Mathematics

Semester

2

Credits

6

Number of Teaching Hours

36

Number of Tutoring Sessions

24

Number of Laboratory Sessions

0

This course is optional

Content

Objective

Content

Hilbert spaces: Inner products, orthogonality, orthogonal projections, orthonormal bases, application to the space L2 and Fourier series. Linear operators: nuclear operators, Hilbert-Schmidt operators, compact operators, bounded operators, unbounded operators. Basic properties. Spectrum and singular values of a class of operators, inequalities between singular values, minimax properties, Schmidt development, elementary properties. Application to differential equations. General theorems: Schauder’s theorem. Riesz theorem. Fredholm alternative. Allahverdiev’s theorem. Spectral decomposition theorem.