# About the adjoint of some differential operator on $L^2(0,1)$

If $$T=-d^2/dx^2$$ is defined on the domain $$D(T)=\{f\in C^2[0,1]: f(1)=f'(0)=f'(1/2)=0\}\subset L^2(0,1)$$. What's the Hilbert adjoint operator $$T^*$$?

• Well, I thought it should be $D(T^*)=\{f\in H^2(0,1): f(1)=f'(0)=0\}$ but in a paper where they treat a similar question , they find $D(T^*)=\{f\in H^2(0,1): f(1)=f'(0)=f'(1/2)=0\}$. – Math Apr 30 at 18:59