$\langle2,1\rangle$-Compact operators

We consider the class of the continuous $L_{2,1}$ linear operators in $L_2$ that are sums of the operators of multiplication by bounded measurable functions and the operators sending the unit ball of $L_2$ into a compact subset of $L_1$. We prove that a functional equation with an operator from $L_{2,1}$ is equivalent to an integral equation with kernel satisfying the Carleman condition. We also prove that if $T\in L_{2,1}$ and $VTV^{-1}\in L_{2,1}$ for all unitary operators $V$ in $L_2$ then $T=\alpha1+C$, where $\alpha$ is a scalar, 1 is the identity operator in $L_2$, and $C$ is a compact operator in $L_2$.