# When does $\lVert T_n \rVert_p \to \lVert T_0 \rVert_p$ imply $\lVert T_n - T_0 \rVert_p \to 0$?

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:

1. $$T_n \to T_0$$ in the strong operator topology
2. The singular values of $$T_0$$ and/or the $$T_n$$ have some property.
• The other hypothesis is $T_n \to T_0$ in some sense. Dec 16, 2019 at 20:58
• @mr_e_man You can not replace $f_0$ by $-f_0$, I should have given more details though. The full statement of the Lemma is: Let $(f_n)$ be a sequence in $L^p$ such that $f_n \to f_0$ almost everywhere. Then $\lVert f_n \rVert_p \to \lVert f \rVert_p$ implies $\lVert f_n - f_0 \rVert_p$. I have edited my question as well, so hopefully it's clear now. Dec 16, 2019 at 22:52
• @GeraldEdgar How do you know that hypotheses of different forms (for example a specific form of the spectrum or the singular values) can be excluded a priori? Dec 16, 2019 at 22:57
• If $x_n\to x$ weakly in a Hilbert space and $\|x_n\|\to\|x\|$, then $\|x_n-x\|\to 0$. This gives such a result for $p=2$. Dec 16, 2019 at 23:00
• In uniformly convex spaces weak convergence of $x_n$ together with $\|x_n\|\to\|x\|$ implies convergence. I am not sure whether this helps. Dec 17, 2019 at 5:37

In the case $$E$$ evn :

if $$\{T_n ;n\in \mathbb N\}$$ compact and $$\text{card}(\{n \in \mathbb N; ||T_n||=||T_0||\})\in\mathbb N$$

then $$||T_n|| \rightarrow ||T_0||$$ imply $$T_n \rightarrow T_0$$

or

$$K$$ compact with $$\forall n \in \mathbb N, T_n\in K$$

if $$\text{card}(\{a \in K; ||a||=||T_0||\})=1$$ then $$||T_n|| \rightarrow ||T_0||$$ imply $$T_n \rightarrow T_0$$

Remark : if $$\{T_n ;n\in \mathbb N\}$$ isn't compact then we have not $$T_n \rightarrow T_0$$

• These conditions seem too restrictive Dec 18, 2019 at 3:54
• @YemonChoi : and now? Dec 18, 2019 at 4:24
• @Dattier Thanks for your answer! I have to agree with Yemon Choi and think that these conditions are quite restrictive and also make no specific use of the Schatten-norms, so I'll keep the question open. Dec 19, 2019 at 8:03
• @r_faszanatas well, it's your choice Dec 19, 2019 at 8:06