One define the operator $T$ as :$$T: = (I - {{{\partial ^2}} \over {\partial {x^2}}}):H_0^1(0,L) \cap {H^2}(0,L) \to {L^2}(0,L) $$ let $f \in H_0^2(0,L) \cap {H^4}(0,L)$. What can we say about ${T^{ - 1}}{{{\partial ^4}f} \over {\partial {x^4}}}$ ? In which space is it?
My attempt is as following : ${T^{ - 1}}:{L^2}(0,L) \to H_0^1(0,L) \cap {H^2}(0,L)$ so ${{{\partial ^4}f} \over {\partial {x^4}}} \in {L^2}(0,L)$ So $${\left\| {{T^{ - 1}}{{{\partial ^4}f} \over {\partial {x^4}}}} \right\|_{{L^2}}} \le c{\left\| {{{{\partial ^2}f} \over {\partial {x^2}}}} \right\|_{{L^2}}}. $$ Is this write ? Thank you.
