Equivalent operator norm

Question

Let $E$ be a Banach space and let $L:E \to E$ be a bounded operator. I want to know when de we have the equivalence between the norms ${\left\| {Lx} \right\|_E}$ and ${\left\| {x} \right\|_E}$. More precisely,what is the condition on $L$ to obtain a such equivalence. Thank you.

Answer 1

If I understood you correctly, you want to find a characterization of bounded linear maps $L:E\to E$ such that there exist constants $c, C > 0$ such that $$ c\Vert x \Vert \leq \Vert Lx \Vert \leq C \Vert x \Vert. $$ Notice that the right inequality is trivial, since $L$ is bounded.

As for the left inequality, it is a nice exercise to show that the following are equivalent:

1. $L$ is bounded below, i.e. there exists a constant $c>0$ such that $c \Vert x\Vert \leq \Vert Lx\Vert$;

2. $L$ is injective and its range is closed;

3. The transpose $L':E'\to E'$ of $L$ is surjective.