Functional Analysis – Finding the Adjoint of an Operator

Question

This is from my homework, I'm totally lost as to how to proceed.
Consider the operator $T: L^2([0,1]) \rightarrow L^2([0,1])$ defined by
$(Tf)(x) = \int^x_0 f(s) \ ds$
What is the adjoint of $T$?

This operator doesn't seem to be an orthogonal projection, nor is it self-adjoint. How does one find the adjoint of an operator in general? Thanks in advance!

Answer 1

Using the fact that

$$ \langle Tf , g \rangle=\langle f , T^{*}g \rangle, $$

we have

$$ \langle Tf, g\rangle = \int_{0}^{1} (Tf)(t)g(t)\,dt =\int_{0}^{1} \int_{0}^{t} f(\tau)\,d \tau\, g(t)\, dt = \int_{0}^{1} f(\tau)\, \left(\int_{\tau}^{1} g(t) \,dt\right)\, d \tau $$ $$ = \langle f, T^{*}g\rangle $$

From the last integral, we can see that the adjoint is given by

$$ (T^{*}f) (x) = \int_{x}^{1} f(s)\, ds $$