Differential operator and permutation

Question

We consider the operator $$T: = \left(1 - \gamma^2 {{{\partial ^2}} \over {\partial {x^2}}}\right):H_0^1(0,L) \cap {H^2}(0,L) \to {L^2}(0,L)$$ is obvious that $T$ is invertible and its iverse defined by :$${T^{ - 1}}:{L^2}(0,L) \to H_0^1(0,L) \cap {H^2}(0,L)$$ Now we consider tyhe operator ${{{\partial ^4}} \over {\partial {x^4}}}{T^{ - 1}}$ and ${T^{ - 1}}{{{\partial ^4}} \over {\partial {x^4}}}$ with same domain $H_0^1(0,L) \cap {H^2}(0,L)$ and their images in $L^2$. My first question: are both operators equal? my second question is :if they are equal can i write $$\left({T^{ - 1}}{{{\partial ^4}} \over {\partial {x^4}}}u,u\right) = \left({{{\partial ^4}} \over {\partial {x^4}}}{T^{ - 1}}u,u\right) = \left({{{\partial ^2}} \over {\partial {x^2}}}{T^{ - 1}}u,{{{\partial ^2}} \over {\partial {x^2}}}u\right) = \\=\left({T^{ - 1}}{{{\partial ^2}} \over {\partial {x^2}}}u,{{{\partial ^2}} \over {\partial {x^2}}}u\right) = \left({T^{ - 1/2}}{{{\partial ^2}} \over {\partial {x^2}}}u,{T^{ - 1/2}}{{{\partial ^2}} \over {\partial {x^2}}}u\right) = \left|{T^{ - 1/2}}{{{\partial ^2}} \over {\partial {x^2}}}u\right|^2$$ where$$u \in H_0^2(0,L) \cap {H^4}(0,L)$$ Thanks.