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You are right, the best constant is $1$. In fact, the stronger inequality $\gamma_2(A) \leq \|A\|_{\infty}$ is also true (and is stronger since $\|A\|_{\infty}\leq \| A\|_{1}$). For simplicity I deno …

No, no such constant exists. For example, if $I = \{(i,j) \mid i<j\}$, then $\Sigma\mapsto \Sigma_I$ is the usual triangular projection, and the norm is of order $\log n$, see for example Norm of the …

Jesse answered your question, but let me correct your last remark. Namely if $A,B$ are positive matrices, $\|AB\|_{tr}$ and $\|BA\|_{tr}$ are more than equivalent, they are equal! In fact more genera …

I expand my comment where I claim that, on the space (call it $N(V)$) of all norms on $V$, the smallest topology making continuous the evaluations $\|\cdot\| \mapsto \|v\|$ (for $v \in V$) coincides w …

The claim is true. More generally the Hölder inequality holds for the Schatten $p$-norms. The statement is here without proof in Wikipedia, and by induction it implies that for matrices $X_1,\dotsc, X …