# Homework 2, Math 224a Fall 2020 ### Updated until Sep 17; due Oct 1 in class. 1. Reed and Simon VI.9a:Prove that if $A=A^*\in \mathcal{L}(H)$ then $$\|A\|=\sup_{\|x\|=1}(x,Ax).$$ Also do 9b: give an example showing that this is not true in general for non self-adjoint operators. 8. Reed and Simon VI.14. 10. Reed and Simon VI.19 11. Reed and Simon VI.41. 12. Prove that if $T_n$ is a Cauchy sequence of compact operators, then $T_n\rightarrow T$ for some compact $T$. You may use the fact that every compact operator is the norm limit of finite rank operators. hint: use a diagonalization (in the sense of logic, not linear algebra) argument. 13. Prove that if $T\ge 0\in K(H)$ with eigenvalues $\lambda_1\ge \lambda_2\ge\ldots,$ then $$\lambda_k = \max_{V:dim(V)=k}\min_{x\in V\setminus\{0\}} (x,Tx)/\|x\|^2.$$ This is half of the Courant-Fisher theorem. 14. **(Cancelled, I did this in class on 9/17)** Prove that if $T$ is trace class then there are Hilbert-Schmidt operators $T_1,T_2$ such that $T=T_1T_2$ (hint: use the SVD). Mimicking the proof in class for positive operators, use this to show that for such $T$, the trace $$\newcommand{\tr}{\mathrm{tr}}\tr_\phi(T)=\sum_n (\phi_n, T \phi_n)$$ is invariant of the choice of ONB $\phi$. Conclude that $\tr(T)=\sum_n \lambda_n$ for all self-adjoint trace class $T$. 15. Give an example of a compact operator which is not trace class. 16. Reed and Simon VI.26