I would like to start by saying that any comment or idea is highly appreciated.

Let us observe that for Hilbert-Schmidt operators $H_1,H_2$ on an infinite-dimensional separable complex Hilbert space $H$ and a bounded self-adjoint operator $T: H \rightarrow H$ $$\operatorname{Tr} \left(H_1 ([T,H_2])^*\right) = \operatorname{Tr} \left([T,H_1] H_2^* \right).$$

In other words, the operator $S_T(H_1):=[T,H_1]$ is self-adjoint with respect to the Hilbert-Schmidt inner product.

Then, we can look at $0 \le S_T^2(H_1) = [T,[T,H]]$ and see that this operator is positive, as it is a square.

Now, I would like to ask the following questions:

(i)Is there a self-adjoint bounded $T$ such that $S_T^2$ has a spectral gap? (FALSE! See answer by Mateusz Wasilewski.)

(ii)Is there a finite collection of self-adjoint bounded $T_1,...,T_n$ such that $S_{T_1}^2 + ... + S_{T_n}^2$ has a spectral gap?

(iii) Is there an infinite collection of positive bounded $(T_n)$ such that $\sum_n T_n$ converges pointwise such that $\sum_n S_{T_n}^2$ has a spectral gap?

I say $S_T^2$ has a spectral gap if $$\operatorname{Tr}(S_T^2(H_1)H_1^*) \ge \lambda \operatorname{Tr}(H_1H_1^*)$$

for some positive $\lambda>0$.

It looks to me that somebody with a operator algebra background may be more familiar with this problem. I assume the three questions are increasingly likely to be true?