Hello. I'm trying to find the adjoint of A, where A is unbounded operator in , D(A)=C[0,1], (Af)t=f(0).
I think, is operator . Is this correct? And what is ?
If , with , then for all f in C[0,1]. Writing the inner products as integrals, . In particular, if f(0) = 0 then . But continuous functions vanishing at 0 are dense in , so if h is orthogonal to all such functions then h = 0.
So I reckon that is the zero operator. Next, what is the domain of ? Well, if and then for all f in C[0,1]. That condition is equivalent to . So it looks as though the domain of is (where 1 denotes the constant function 1 on [0,1]).