Example of bounded form below unbounded linear operator

Question

We say that an unbounded linear operator $A$ on a Banach space $X$ is bounded from below (or just semibounded), if there exists a constant $c>0$ such that for all $x\in D(A)$ we have $$ \|Ax\| \geq c\|x\|. $$ My question: is there any concrete examples in general Banach spaces (not Hilbert spaces) of this kind of (unbounded) operator? Thank you.

Answer 1

Let $A$ act on $\ell^1(\mathbb{N})$ by $$(Ax)_n= nx_n$$ with domain $D(A)$ consisting of the sequences with finitely many nonzero terms. Then $\|Ax\|_1\ge\|x\|_1$.