# Reverse statement of Kato-Rellich Theorem

The Kato-Rellich Theorem states: If $$T$$ is a self-adjoint operator in the hilbert space $$\mathcal{H}$$ and if $$A$$ is symmetric and $$T-\text{bounded}$$ with $$T$$-bound $$< 1$$. Then $$T+A$$ is self-adjoint. I´m wondering if the following is true:

$$\text{If}~~T~~\text{is s.a. in}~~\mathcal{H}~~\text{and if}~~A~~\text{is symmetric and if it holds that}~~\overline{T+A} = \left(T+A\right)^{*},~~\text{then}~~A~~\text{is}~~T-\text{bounded}.$$

A counterexample is the quartic quantum anharmonic oscillator $$H+V$$ with $$V=x^4$$. On the Hermite function basis $$\{e_n\}$$, $$He_n = ne_n$$ while $$\vert Ve_n \vert \gt$$ const.$$\times n^2$$. The latter is obvious from the creation/annihilation representation. There are various proofs of e.s.a. for $$H+V$$.