# Relative boundedness of the adjoint

Let $X$ be a separable Banach space and $T_1:D(T_1) \subset X \rightarrow X$ and $T_2:D(T_2) \subset X \rightarrow X$ two closed operators with $D(T_2)\subset D(T_1)$ and $D(T_2^*) \subset D(T_1^*).$

We say that $T_1$ is relatively $T_2$ bounded if $D(T_2) \subset D(T_1)$ and for all $x \in D(T_2)$

$$\left\lVert T_1x \right\rVert \le \alpha \left\lVert T_2 x \right\rVert + \beta \left\lVert x \right\rVert.$$

I am interested in the following question:

Are there sufficient conditions such that $T_1^*$ being relatively $T_2^*$ bounded implies that $T_1$ is $T_2$ bounded with the same relative bound?

What about the converse: Does $T_1$ being relatively $T_2$ bounded imply that $T_1^*$ is relatively $T_2^*$ bounded under the above conditions?

• Are $\alpha$ and $\beta$ positive numbers? – Meisam Soleimani Malekan Jul 16 '18 at 20:36
• It would be interesting to have a little context for this: where is this definition from? What known examples do you have where things work as you hope? What is the potential application (if it's not too hard / off-topic to explain...) – Matthew Daws Jul 17 '18 at 7:44