Let $L$ be some linear and compact operator on some Hilbert space. And denote by $L^*$ the adjoint operator.
I need to bound for some $c>0$:
$$ \left\|(L^*L+c I)^{-\frac{1}{2}}L^*\right\|\leq1 $$ what i tried was
$$ \left\|(L^*L+c I)^{-\frac{1}{2}}L^*\right\|^2= \left\|L(L^*L+c I)^{-1}L^*\right\| $$
However from here I get stuck. I think I need some functional calcul analysis to understand how this works ? I would understand it if
$$ \left\|L(L^*L+c I)^{-1}L^*\right\|\,\, (?=?)\,\, \left\|L^*L(L^*L+c I)^{-1}\right\| = \sup_{\sigma\in r(L^*L)}\left\|\sigma(\sigma+c)^{-1}\right\|\leq1 $$ However in general the operators do not commute under the operator norm or do they ? I would be very glad if someone could tell me what I am missing here.
What you need to use is the C$^*$-identity $\|T^*T\|=\|T\|^2=\|TT^*\|$. So $$ \|(L^*L+cI)^{-1/2}L^*\|^2=\|(L^*L+cI)^{-1/2}L^*L(L^*L+cI)^{-1/2}\| =\|L^*L(L^*L+cI)^{-1}\|\leq1. $$ Once you get an $L^*L$ things do commute. And now $L^*L$ is positive, so you can use functional calculus, the spectral theorem, or other techniques to show that last inequality.