Let $K$ be a continuous linear operator on $C[0,1]$ (more, precisely, it is a linear integral operator). Then $K$ defines a continous linear operator $\widehat K$ on $\mathcal L(C[0,1])$ by the rule $$ \widehat KA=KAK, \quad A \in \mathcal L(C[0,1]), $$ with norm $\|\widehat K\| \leq \|K\|^2$. My question is what is a reasonable framework to study properties of $\widehat K$ in terms of operator $K$? I recall that $K$ is the integral operator so, for example, I would like to be able to formulate conditions of invertibility of $\mathrm{Id} - \widehat K$ in terms of integral operator $K$.
In the case of matrices, operator $\widehat K$ corresponds to Kronecker product of matrices $K^T \otimes K$ so that invertibility of operator $\mathrm{Id}-\widehat K$ corresponds to invertibility of matrix $\mathrm{Id} - K^T \otimes K$. Is it possible to find some analogue of this contruction in the case of operators on $C[0,1]$? I know that in the case of Banach spaces there are many ways to naturally define the tensor product, so I'm not sure that the above construction generalizes well.
