Compute the adjoint of an operator in Hilbert space

Question

I have the following question. In the Hilbert space $l^2$, consider the operator $Tx=(\frac{x_n+x_{n+1}}{2})_n$ and $x=(x_n)_n$. Compute the adjoint of operator $T$.

I tried to find $T^*$ such that $(Tx,y)=(x,T^*y)$ through representing $x$ in series such that $x=\Sigma_{k=1}^\infty(x,e_k)e_k $. Then, I'm stuck.

Answer 1

Here are two ways of solving this problem

Hint 1: View $T$ as an infinite matrix

$$\begin{pmatrix}1/2 & 0 & 0 & \dots \\1/2 & 1/2 & 0 & \dots \\ 0 & 1/2 & 1/2 & \dots \\ \vdots & \vdots & \vdots & \ddots\end{pmatrix}$$

What would you get when you calculate the adjoint as if it was a finite matrix?

Hint 2: Let $S$ be a right shift operator on $l^2$. Then $T = 1/2(S+1)$ where $1$ is the identity operator. Then $$T^* = 1/2 (S^* + 1^*) = 1/2(S^* + 1)$$

What is the adjoint operator of a right shift operator? (Your intuitive guess is correct).