This is a math question, not a CVX question. Nevertheless, …
I will assume X \succeq 0, but do no need to assume anything about rank(X)
Let M = H*X*H'. Then M is psd.
Therefore max(eig(M)) <= trace(M). Therefore (56) implies(57).
However, (57) does not imply(56).
ok，thanks, i have got it.