Let $T$ be a linear densely defined operator on a Hilbert space $H$ and $L$ be a selfadjoint operator with discrete spectrum such that $L^{-1}$ is bounded and $$\|Tf\| \leq \Phi_{\eta}(f) + \eta \|L^{a}f\|, \quad\forall~f \in \mathcal{D}(L),$$ where $0< a<1$, $\Phi_{\eta}$ is a continuous convex functional and $\eta$ is a small positive constant. I think to write $$\|L^{-a}Tf\|\leq L^{-a} \Phi_{\eta}(f) + \eta \|f\|$$
But I want know if $ L^{-a} \Phi_{\eta}(f)\leq C \|f\| $? (where $C$ is a positive constant.)
In other words I want know if $L^{-a}T$ is bounded?