As I was solving exercise E.3.4.2. in Gert Pedersen's Analysis Now, I got two questions.
1. Let $1 \leq p < \infty$. Pedersen defines Schatten $p$-ideals $\mathcal{B}^p (\mathcal{H})$ as the set of compact operators $T \in \mathcal{B}_0 (\mathcal{H})$ such that $|T|^p$ is trace class. He then defines the norm on the Schatten ideals: $$\|T\|_p = (\mathrm{tr} (|T|^p))^{\frac{1}{p}}.$$ The exercise asks us to prove that the Schatten ideals are in fact ideals and the $p$-norm really is a norm. His hint for proving the additivity (and the triangle inequality) is essentially to consider $S, T$ and then take the polar decompositions $$S = U |S|,\quad T = V|T|,\quad S + T = W|S + T|.$$ Then we are supposed to take any finite projection $Q$ that commutes with $|A + B|$ to get $$\mathrm{tr} (|S + T|^p Q) \leq \mathrm{tr} (|S||S + T|^{p - 1} Q) + \mathrm{tr} (|T||S + T|^{p - 1} Q).$$ Then we are supposed to use the inequality $$\mathrm{tr} (|AB|) \leq (\mathrm{tr} (S^p))^{\frac{1}{p}} (\mathrm{tr} (S^q))^{\frac{1}{q}}$$ for positive $A, B \in \mathcal{B}^p (\mathcal{H})$ and $p^{-1} + q^{-1} = 1$. According to the author, we can use this inequality to get the bound $$\mathrm{tr} (|S||S + T|^{p - 1} Q) \leq \|S\|_p \| |S + T| Q \|_p ^{p - 1}$$ and similarly $$\mathrm{tr} (|T||S + T|^{p - 1} Q) \leq \|T\|_p \| |S + T| Q \|_p ^{p - 1}.$$ It is this last step that I'm having trouble with: why is $\mathrm{tr} (|S||S + T|^{p - 1} Q) = \mathrm{tr} (\left||S||S + T|^{p - 1} Q\right|)$?
2. Wikipedia claims that if we take all bounded (not necessarily compact) operators $S \in \mathcal{B} (\mathcal{H})$ such that $|S|^p$ is trace class, then this is a Banach space with the above norm. Bizarrely enough, I couldn't find one reference on the internet that proved this for a general non-separable Hilbert space. Is there perhaps an elementary argument that generalized Pedersen's proof to non-compact operators?
Thanks in advance!