help,example about disjoint operators

Question

$T\colon L^2[0,1]→L^2[0,1]$ is given by $$ Tx(t)=∫_0^1 tx(s)\,ds $$ How can we find adjoint operator of $T$ in this space? $\langle Tx,y\rangle= \langle x,T^*y\rangle$ should be okay.But what are we take for $y$ and how is continue?

Answer 1

$\langle Tx, y \rangle = \int_0^1 Tx(t) \cdot y(t) dt = \int_0^1 \int_0^1 t \cdot x(s) ds y(t) dt = \int_0^1 \int_0^1 t \cdot y(t) dt x(s) ds = \langle???, x\rangle$

Where the change of the order of integration is justified by Fubini's theorem.

This should allow you to calculate $T^{\ast}y$ for $y \in L^2([0,1])$.

EDIT: You don't make any specific choice for $y$. You just use the defining equation $\langle Tx, y\rangle = \langle x, T^{\ast} y\rangle$ of the adjoint (not disjoint) operator to calculate $T^{\ast}y$.