Let || . ||_p denote the Schatten p-norm of a matrix of complex numbers. Does it hold that ||AB||_r ≤ ||A||_p ||B||_q for all matrices A,B and all 1 ≤ p,q,r ≤ ∞ with 1/p + 1/q = 1/r?

Wikipedia (https://en.wikipedia.org/wiki/Schatten_norm#Properties) says yes and gives two citations, but I'm not convinced by either of them:

• Eq. (4) and Theorem 4.2(i) of https://msp.org/pjm/1986/123-2/pjm-v123-n2-p03-s.pdf seem to state the inequality, except that if I understand the definition at the top of Page 271 correctly, the authors require A and B to be PSD. And even if the general case of the inequality reduces to this special case, I lack the background in (I think?) functional analysis to parse the definition at the top of Page 271 and confirm that it generalizes the Schatten p-norm of a matrix.

• https://link.springer.com/article/10.1007/BF01231769#preview doesn't seem to state anything like the inequality.

If the inequality is true, I'm looking for either of the following:

• A citable reference for the inequality. If, like Wikipedia's references, the reference you give proves a more general statement, then you may need to explain how the inequality I'm looking for is a special case of that statement.

• A simple proof, if one exists. Note that I'm mainly interested in the r>1 case.

If the inequality is false or an open problem, I'm looking for a convincing argument that this is so.