3
$\begingroup$

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.

$\endgroup$
2
$\begingroup$

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).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.