maybe this question is too elementary but I could not find any results in the functional analysis textbooks I own:

For separable Hilbert spaces $H$, $K$, let $S_p(H,K)$ be the $p$th Schatten class of linear operators $T \colon H \to K$. I'm interested in conditions which imply that for a sequence $(T_n)_{n \in \mathbb{N}_0} \subseteq S_p(H,K)$ the implication

$$\lVert T_n \rVert_p \to \lVert T_0 \rVert_p \Rightarrow \lVert T_n - T_0 \rVert_p \to 0$$

is satisfied.

In finite dimension, a related result states that for $p>0$ and a sequence $(f_n) \subseteq L^p(\mathbb{R},\mu)$ which satisfies $f_n \to f_0$ almost surely, one has that $\lVert f_n \rVert_2 \to \lVert f_0 \rVert_p$ implies $\lVert f_n - f_0 \rVert_p \to 0$ (see for example Lemma 1.32 in Kallenberg's *Foundations of modern probability*). Unfortunately the proof does not seem to be transferable but gives as a heuristic that the implication holds in $S_p$ if convergence in the strong operator topology is assumed in addition.

**UPDATE:**

Thanks to everybody for the comments! Apparently my question is a bit too general (I asked it out of pure interest with no specific research goal in mind), so let me be more specific:

Does the above implication hold if we assume any or some of:

- $T_n \to T_0$ in the strong operator topology
- The singular values of $T_0$ and/or the $T_n$ have some property.

2more comments