# Self-Adjointness of sum of $\ell^{2}$ operators using Kato Rellich Theorem

Consider the two operators on $$\ell^{2}$$: $$T\left(x_{n}\right)_{n} = \left(nx_{n}\right)_{n}$$ and $$V\left(x_{n}\right)_{n} = \left(\sqrt{n}x_{n}\right)_{n}$$ defined on $$\text{dom}(T) = \left\{x \in \ell^{2} : (nx_{n})\in \ell^{2}\right\}$$ and $$\text{dom}(V)= \left\{x \in \ell^{2} : (\sqrt{n}x_{n})\in \ell^{2}\right\}$$ respectively. I want to show that $$T+V$$ is self-adjoint using Kato-Rellich-Theorem. I could show that $$T$$ is s.a and $$V$$ is symmetric. However I run into troubles in showing that $$V$$ is $$T$$-bounded, with bound $$<1$$ i.e. that there are $$a>0$$ and $$b \in (0,1)$$ such that: $$\|Vx\|^{2}\le a\|x\|^{2} + b\|Tx\|^{2}$$. Can someone give me a hint on how to get this inequality? $$\|Vx\|^{2} = \sum_{n}nx_{n}^2 \le ??$$ How can I split this up into two terms as above? Any ideas or help is greatly appreciated!

We have $$|| V x ||^2 = \sum_{n} n x_n^2 \le \sum_n \left( \frac{1}{2} + \frac{n^2}{2}\right) x_n^2 = \sum_n \frac{n^2}{2} x_n^2 + \sum_n \frac{1}{2} x_n^2 = \frac{1}{2} \| T x \|^2 + \frac{1}{2} \| x \|.$$ Note that also all values of $$a > \frac{1}{2}$$ are possible.