Given the operator $\displaystyle T: L^2([0,1]) \rightarrow L^2([0,1]) $ defined by
$\displaystyle (Tf)(x) = \int^x_0 f(t) \ dt$
How do I find adjoint of $\displaystyle T$? I'm totally lost as to how to go about doing this. Thanks in advance!
Given the operator $\displaystyle T: L^2([0,1]) \rightarrow L^2([0,1]) $ defined by
$\displaystyle (Tf)(x) = \int^x_0 f(t) \ dt$
How do I find adjoint of $\displaystyle T$? I'm totally lost as to how to go about doing this. Thanks in advance!
In an inner product space, the "adjoint" of a linear transformation, A, is the linear transformation A* such that <Au, v>= <u, A*v>. Here the inner product is $\displaystyle <u(x), v(x)>= \int_0^1 u(x)v(x)dx$. So you are looking for A* such that $\displaystyle <Au, v>= \int_0^1 \left(\int_0^x u(t)dt\right)v(x)dx= \int_0^1 u(x)A*(v(x))dx$. Do the integrals on the left and find the function A* that makes the right side equal to that. I would suggest using "integration by parts".
You actually obtain
$\displaystyle \int_0^1 U(x)v(x)dx = U(x)V(x)|_0^1-\int_0^1u(x)V(x)dx$
That is
$\displaystyle \langle u, T^*v\rangle = \langle Tu,v\rangle = U(x)V(x)|_0^1 - \langle u,Tv\rangle$
This suggest that you need to write the product $\displaystyle U(x)V(x)$ in a form with $\displaystyle u$ in it. That is, you need to make it look more like $\displaystyle \langle u,Tv\rangle$ and somehow combine the two. I hope this helps.