Sequence of increasing null spaces and range Find a vector space W and T ∈ L(W)  such that null T^k ⊊ null T^(k+1) and range T^k ⊋ range T^(k+1) for every positive integer k.

Let $E$ an infinite dimensional vector space, with a basis $\{e_0,e_1, \cdots \}$. Define $T$ by $T(e_0)=0$, $T(e_{2k})=e_{2(k-1)}$ for $k\geq 1$, and $T(e_{2m+1})=e_{2m+3}$ if $m\geq 0$. We have ${\rm Ker}(T)={\rm Span}\{e_0\}=<e_0>$, and ${\rm Range}(T)=< e_{2k}, k\geq 0, e_{2l+1}, l\geq 1>$. We see ${\rm Ker}(T^2)=<e_0, e_2>$ and ${\rm Range}(T^2)=< e_{2k}, k\geq 0, e_{2l+1}, l\geq 2>$. Now generalise this and find ${\rm Ker}(T^k)$ and ${\rm Range}(T^k)$ for $k\geq 2$.  
