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The definition is given for operators between
Hilbert spaces. Therefore speaking of "eigenvalues of the Hermitian matrix" does not make sense. I suggest to speak only about compact operators such that "Hermitian matrix" can be replaced by "non-negative compact self-adjoint operator" here.
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One option would be to restrict the discussion to finite-dimensional
Hilbert spaces first, and then extend it to infinite-dimensional Hilbert spaces and introduce the Schatten class operators.
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The article claims that this is proved in "Sharp uniform convexity and smoothness inequalities for trace norms", but I looked at the paper and couldn't find the claim.
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Confusion about citation for |ST|_r \le |S|_p |R|_q.
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https://en.wikipedia.org/Matrix_norm#Schatten_norms
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