Since I don't have a proper academic training in functional analysis, please forgive any error below or abuse of notation.
Let $\Omega=(0,1)$ be the unit open interval. Let $H^k(\Omega) = W^{k,2}(\Omega, \mathbb R)$ denote the Sobolev space of order $k$. Let $D: H^k(\Omega) \to H^{k-1}(\Omega)$, $k>0$ be the differential operator defined by $D u = u', \forall u \in H^k(\Omega)$. Finally let $B_\alpha : H^2(\Omega) \to L_2(\Omega)$, $\alpha \in \bar \Omega$ be defined by $B_\alpha u = u'(\alpha)$.
I would like to construct a deflated complimentary operator $\bar D: H^2(\Omega) \to L_2(\Omega)$ such that $\text{im}(\bar D) = \text{im}(D) / \text{im}(B_\alpha)$ so that $\text{im}(D) = \text{im}(\bar D) \oplus \text{im}(B_\alpha)$.
How can I build such an operator? and make its definition explicit?