Trace-norm of integral operator

Question

Let me start by saying that I do appreciate any insight on this. So also if you have a partial result, please share it as a comment or answer.

This is somewhat unrelated to what I normally do, so I may be missing something rather obvious here, but unlike for Hilbert-Schmidt norms, very little useful methods seem to be available to calculate the norm of Trace-class operators.

Let $f \in C_c^{\infty}(\mathbb{R}\times\mathbb{R}),$ then we have an integral operator

$$Tg(s):=\int_{\mathbb{R}^2} f(s+t_1,t_2) g(t_1,t_2)dt$$ where $T:L^2(\mathbb{R}^2) \rightarrow L^2(\mathbb{R}).$

My goal is to compute its trace-norm or get at least some useful bounds on it. For this, it may be good to have its adjoint at hand

$$T^*h(t_1,t_2):= \int_{\mathbb{R}}\overline{f(s+t_1,t_2)} h(s)ds.$$

However, now it is still completely unobvious to me how to compute $\sqrt{T^*T}$ which is needed in the standard definition of the trace norm(=nuclear norm).

Possible ways to calculate the nuclear norm that I could imagine to be useful include

a.) Use $\left\lVert T \right\rVert_{\text{nuclear}}= \sum_{n} \left\lvert \langle \sqrt{T^*T}e_n,e_n \rangle \right\rvert$ for an arbitrary orthonormal basis $(e_n)$

b.) Use Hahn-Banach, i.e. $\left\lVert T \right\rVert_{\text{nuclear}} = \sup_{S \in L(L^2(\mathbb{R}),L^2(\mathbb{R}^2)); \left\lVert S \right\rVert=1 } \operatorname{tr} (T^*S)$ or

c.) Use that $\left\lVert T \right\rVert_{\text{nuclear}}$ is the supremum of all $\sum_{n} \lvert\left\langle e_n,Tf_n \right\rangle \rvert$ for ONS $(e_n),(f_n).$ (this may be useful to get bounds)

Questions: I am paticularly curious to find out whether:

1.) There are any theorems or tricks that apply to this operator which allow me to compute its nuclear norm.

2.) Just based on intuition I would assume that the nuclear norm would be something like $$ \int_{\mathbb{R}^2} \left\lvert f(t_1+t_1,t_2) \right\rvert dt_1 dt_2 = \frac{1}{2} \left\lVert f \right\rVert_{L^1}.$$ Can we say if this is at least a correct lower/upper bound for the trace-norm? Maybe it is possible to find an operator $S$ or orthonormal bases (in the definition of the nuclear norms above) so that we get this as a lower bound, at least?

3.) Are there any non-trivial upper/lower bounds available?

If you have any questions about this, please let me know.

Answer 1

This is typically not a trace class operator. The problem is that the kernel $$ K(t,u) = \int \overline{f(s+t_1,t_2)}f(s+u_1,u_2)\, ds $$ of $T^*T$ depends on the first coordinates only through the difference $t_1-u_1$, so has no uniform decay in these directions. It follows that $\int\!\!\int |K|^2\, dt\,du =\infty$ (for typical $f$ at least), so $T^*T$ is not Hilbert-Schmidt and thus $(T^*T)^{1/2}\notin B_4$ (let alone trace class).

In fact, I think $T$ is usually unbounded, for similar reasons (or we can think of $T$ as convolution by $f$ in $\mathbb R^2$, followed by restriction to $\mathbb R \times \{ 0\}$, and this final operation is not bounded).