This is about, a question I answered. Now there is an additional question that I cannot answer and do not want to spend any more time on. I feel like the question will not get any attention, as I already answered it.

Which is the adjoint of $$T\colon\ell^2\to\ell^2, (a_n)_{n\in\mathbb{N}}\mapsto\left(\frac{a_n+a_{n+1}}{2}\right)_{n\in\mathbb{N}}$$

Feel free to answer here: Show that operator is continuous and selfadjoint (or not)

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I assume you talk about $\ell^2(\mathbb{N})$, although it does not make a big difference.

Method 1: compute the matrix of $T$ in the canonical basis of $\ell^2$. You'll find a diagonal of $1/2$, with $1/2$'s right above. Then take the transconjugate of this matrix: this will give you $T^*$.

Method 2: write $T=(Id+S)/2$ where $S$ denotes the unilateral shift $(a_0,a_1,\ldots )\longmapsto (a_1,a_2,\ldots$. Then $T^*=(Id+S^*)/2$, where $S^*:(a_0,a_1,\ldots)\longmapsto (0,a_0,a_1,\ldots)$.

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