Paper 4, Section II, F

Linear Analysis
Part II, 2017

Let HH be a complex Hilbert space with inner product (,)(\cdot, \cdot) and let T:HHT: H \rightarrow H be a bounded linear map.

(i) Define the spectrum σ(T)\sigma(T), the point spectrum σp(T)\sigma_{p}(T), the continuous spectrum σc(T)\sigma_{c}(T), and the residual spectrum σr(T)\sigma_{r}(T).

(ii) Show that TTT^{*} T is self-adjoint and that σ(TT)[0,)\sigma\left(T^{*} T\right) \subset[0, \infty). Show that if TT is compact then so is TTT^{*} T.

(iii) Assume that TT is compact. Prove that TT has a singular value decomposition: for N<N<\infty or N=N=\infty, there exist orthonormal systems (ui)i=1NH\left(u_{i}\right)_{i=1}^{N} \subset H and (vi)i=1NH\left(v_{i}\right)_{i=1}^{N} \subset H and (λi)i=1N[0,)\left(\lambda_{i}\right)_{i=1}^{N} \subset[0, \infty) such that, for any xHx \in H,

Tx=i=1Nλi(ui,x)viT x=\sum_{i=1}^{N} \lambda_{i}\left(u_{i}, x\right) v_{i}

[You may use the spectral theorem for compact self-adjoint linear operators.]