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

Question

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!

Answer 1

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.