Bounding $L^2$ operator norm of integral kernel

Question

Say I have an integral operator $T$ with $g\in L^2$ such that $$\int |Tg(\xi)|^2d\xi=\int g(x)\overline{g(y)}K_1(x,y)\,dxdy.$$ Is it correct that then, by repeated Cauchy-Schwarz, we have that $$\| Tg\|_{L^2}\le \|g\|_{L^2} \|K_1\|^{1/2}_{L^2}?$$