Suppose that I have an infinite matrix $K(i,j)$ with the promise that there is some $C$ such that $\vert K(i,j) \vert \leq C$ for all $i,j \in \mathbb{Z}$.
Consider the Hilbert space $l^2( \mathbb{Z})$ and the trace-class operators on that Hilbert space. For operators $ \rho = \sum_{i,j} \rho(i,j) \vert i \rangle \langle j \vert $ which is trace class. Then consider the operation $S_K(\rho) = \sum_{i,j} K(i,j) \rho(i,j) \vert i \rangle \langle j \vert $. Is $S_K( \rho)$ nescessarily trace class? If yes is there a bound on $\vert \vert S_K \vert \vert_{1,1}$?
The answer and question here is very related: https://mathoverflow.net/questions/406705/if-i-multiply-the-coefficients-of-a-trace-class-operator-with-bounded-complex-nu