In $H = l^2(\mathbb{N})$ equipped with the inner product $\langle u, v \rangle = \sum_{n=0}^{+\infty} u_n v_n$, with $S$ the left shift $(Su)_n = u_{n-1}$ if $n \geq 1$, $(Su)_0 = 0$, and $V$ the right shift $(Vu)_n = u_{n+1}$, and $$\mathcal{L}^2(H) = \left\{T \in \mathcal{L}(H), \quad \|T\|_2 = \sum_{i=0}^{+\infty} \|T(b_i)\|^2 < +\infty\right\}$$ Show that the operators $S$ and $V$ defined in part 2 are not in $\mathcal{L}^2(H)$. Give an example of a non-zero operator in $\mathcal{L}^2(H)$.
In $H = l^2(\mathbb{N})$ equipped with the inner product $\langle u, v \rangle = \sum_{n=0}^{+\infty} u_n v_n$, with $S$ the left shift $(Su)_n = u_{n-1}$ if $n \geq 1$, $(Su)_0 = 0$, and $V$ the right shift $(Vu)_n = u_{n+1}$, and
$$\mathcal{L}^2(H) = \left\{T \in \mathcal{L}(H), \quad \|T\|_2 = \sum_{i=0}^{+\infty} \|T(b_i)\|^2 < +\infty\right\}$$
Show that the operators $S$ and $V$ defined in part 2 are not in $\mathcal{L}^2(H)$. Give an example of a non-zero operator in $\mathcal{L}^2(H)$.