MA6150 BASIC OPERATOR THEORY


Course Details

Dual space considerations: Representation of duals of the spaces c_{00} with p-norms, c_0 and c with supremum-norm, l-p, C[a, b] and L^p. Reflexivity; Weak and weak* convergences.
Operators on Banach and Hilbert spaces: Compact operators between normed linear spaces; Integral operators as compact operators; Adjoint of operators between Hilbert spaces; Self-adjoint, normal and unitary operators; Numerical range and numerical radius; Hilbert--Schmidt operators.
Spectral results for Banach and Hilbert space operators: Eigen spectrum, approximate eigen spectrum; Spectrum and resolvent; Spectral radius formula; Spectral mapping theorem; Riesz-Schauder theory; Spectral results for normal, self-adjoint and unitary operators; Functions of self-adjoint operators.
Spectral representation of operators: Spectral theorem and singular value representation for compact self-adjoint operators; Spectral theorem for self-adjoint operators.

Course References:

Books:
1. T. Nair: Functional Analysis: A First Course, Wiley Eastern, 1981.
2. B.V. Limaye: Functional Analysis, Second Edition, New Age International, 1996.