In $H = l^2(\mathbb{N})$ equipped with the inner product $\langle u, v \rangle = \sum_{n=0}^{+\infty} u_n v_n$, let $T$ be an operator on $H$. We admit the existence of an operator $\tilde{T} \in \mathcal{L}(H)$ such that $$\forall (x,y) \in H^2, \quad \langle T(x), y \rangle = \langle x, \tilde{T}(y) \rangle$$ Let $B = (b_i)_{i \in \mathbb{N}}$ and $C = (c_i)_{i \in \mathbb{N}}$ be two Hilbert bases of $H$ such that $$\sum_{i=0}^{+\infty} \|T(b_i)\|^2 < +\infty$$ Show that $$\sum_{i=0}^{+\infty} \|T(b_i)\|^2 = \sum_{i=0}^{+\infty} \|\tilde{T}(c_i)\|^2$$
In $H = l^2(\mathbb{N})$ equipped with the inner product $\langle u, v \rangle = \sum_{n=0}^{+\infty} u_n v_n$, let $T$ be an operator on $H$. We admit the existence of an operator $\tilde{T} \in \mathcal{L}(H)$ such that
$$\forall (x,y) \in H^2, \quad \langle T(x), y \rangle = \langle x, \tilde{T}(y) \rangle$$
Let $B = (b_i)_{i \in \mathbb{N}}$ and $C = (c_i)_{i \in \mathbb{N}}$ be two Hilbert bases of $H$ such that
$$\sum_{i=0}^{+\infty} \|T(b_i)\|^2 < +\infty$$
Show that
$$\sum_{i=0}^{+\infty} \|T(b_i)\|^2 = \sum_{i=0}^{+\infty} \|\tilde{T}(c_i)\|^2$$