I was wondering if there is an analogue to the classical Riesz Thorin theorem for Schatten classes. I suppose the answer is yes, since Schatten classes are so similar to $\ell^p$ spaces for which the theorem holds. However, I could not find a reference online. In particular, does one have to take the bounded or compact operators as the infinity space when interpolating?- I suppose bounded ones as they are the dual of the trace-class operators which fits to $(L^1 )'=L^{\infty}.$

A Riesz-Thorin interpolation theorem (and a Marcinkiewicz one) is known to hold for Schatten classes (with the case $p=\infty$ corresponding to the *compact* operators, think of the canonical duality $c_0' = \ell^1$): cf. [1, Thm. 13.1], [2, Thms. 2.9-10] and [2, Remark 1 on p. 23].

*References*

[1] I.C. Gohberg, M.G. Kreīn, Introduction to the Theory of Linear Nonselfadjoint Operators, Amer. Math. Soc., Providence, 1969.

[2] B. Simon, Trace Ideals and Their Applications, 2nd ed., Amer. Math. Soc., Providence, 2005.