Let A be a self-adjoint, compact operator on a Hilbert space. Prove that there are positive operators P and N such that A = P − N and P N = 0.

Question

I'm having trouble approaching this problem. I'm totally unsure how to approach this problem.

Here's what I've tried so far: If A is self-adjoint, then $(A)^*=A^*$ and $(AB)^* = B^* A^*$.

Answer 1

Hint: With the spectral theorem, we can write $$ A(x) = \sum_{k=1}^\infty \lambda_k \langle x,\xi_k \rangle \xi_k $$ where $(\xi_k)_{k \in \Bbb N}$ is an orthonormal sequence (of eigenvectors of $A$) in $H$ and each $\lambda_k$ is real. With that, define $$ P(x) = \sum_{k=1}^\infty \max\{\lambda_k,0\} \langle x,\xi_k \rangle \xi_k $$

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If you would like to avoid using the spectral theorem, then you could also characterize $P$ as $P = \frac 12 (A + \sqrt{A^2})$.