Relationship between the trace distance and the operator norm for some time dependent integral operator with kernel $K(x,y,t)$.

Question

Let us assume that $L_{K_{t}}$ is a positive, compact operator with time dependent kernel $K(x,y,t)$.

I know that for self adjoint operators

$$ \|A\| = max_{n} |\lambda_{n}|$$

where $\lambda_{n}$ are eigen values (point spectrum elements of $A$).

I also know that for the trace distance $\| A\|_{1} := Tr\{A^{\dagger}A\}$.

$$\| A\|_{1} = \sum_{n}|\lambda_{n}|$$

My question is the following.

Let us assume that assymptotically $L_{K_{t}}$ has zero operator norm. i.e. $$\lim_{t\rightarrow \infty} \|L_{K_{t}}\| = 0 $$

does this not imply that

$$\lim_{t \rightarrow \infty} max_{n}|\lambda_{n}(t)| = 0 $$

and consequently that

$$ \lim_{t\rightarrow \infty} \sum_{n}|\lambda_{n}(t)| =0$$

Answer 1

The conclusion is not true. Consider $\ell^2(\mathbb{N})$ and $t\ge 1.$ $$K(i,j,t)= \begin{cases} t^{-1/2} & i=j\le t \\ 0 & {\rm otherwise}\end{cases} $$ Then $$\|L_{K_t}\|=t^{-1/2},\qquad {\rm tr}\,L_{K_t}=t^{-1/2}[t]$$ Thus the norms tend to $0,$ while the trace norms tend to $\infty.$