Operator Theory 

offered by IIT Hyderabad

Review of Hilbert space Theory , Bounded operators on Hilbert spaces, examples

Adjoint an operator, examples, Self-adjoint, normal, positive, unitary, isometries, partial isometries

Orthogonal projections with examples, invariant subspaces, numerical range and characterization of operators

Banach Algebras, inertibility, spectrum

Gelfand-Mazur theorem, spectral radius formula, spectral mapping theorem.

Subdivion of the spectrum of an operator, properties of the various spectra

Computing spectrum with examples

Existence of square root, polar-decomposition.

Compact operators, properties

Spectral theorem for compact self-adjoint operators, spectral theorem for compact normal operators, Schmidt-decomposition, Monotone convergence theorem for self-adjoint operators

Spectral theorem for self-adjoint operators, continuous functional calculus, spectral theorem for self-adjoint operators( multiplication form)

Spectral theorem for normal operators (both integral and multiplication form), continuous functional calculus for normal operators.