Operator not bounded below on sum of subspaces

Question

Let $X$ be a Banach space. Say that a bounded linear operator $T\colon X\to X$ is bounded below by $\delta>0$ on $Y\subset X$ if $\|Tx\|\geqslant \delta \|x\|$ for all $x\in Y$. Is there a Banach space $X$, an operator $T\colon X\to X$ which is bounded below by $\delta$ on linear subspaces $Y_1, Y_2\subset X$ but not on $Y_1+Y_2$?

Answer 1

Yes. For example, in $\ell_1$ let $Y_1$ be the subspace consisting of elements whose even coordinates are zero and $Y_2$ be the subspace consisting of elements whose odd coordinates are zero. Take $Tx=(x_2-x_1, x_4-x_3, x_6-x_5,\ldots)$. $T$ is an isometry on both $Y_1$ and $Y_2$ but $T(1/2,1/2,1/2^2,1/2^2,1/2^3,1/2^3,\ldots)=0$.