Let H be a complex Hilbert space with inner product (⋅,⋅) and let T:H→H be a bounded linear map.
(i) Define the spectrum σ(T), the point spectrum σp(T), the continuous spectrum σc(T), and the residual spectrum σr(T).
(ii) Show that T∗T is self-adjoint and that σ(T∗T)⊂[0,∞). Show that if T is compact then so is T∗T.
(iii) Assume that T is compact. Prove that T has a singular value decomposition: for N<∞ or N=∞, there exist orthonormal systems (ui)i=1N⊂H and (vi)i=1N⊂H and (λi)i=1N⊂[0,∞) such that, for any x∈H,
Tx=i=1∑Nλi(ui,x)vi
[You may use the spectral theorem for compact self-adjoint linear operators.]