Let $B$ be a separable Banach space, $L:B \to B$ be a hypercyclic operator, $k>0$, $I\_B$ the identity on $B$, and define $L\_k: =k (I\_B - L)$. When is $L\_k$ hypercyclic on $B$? Can anything else be said about $L\_k$?



Difference of hypercyclic operator and identity
I don't know if anything can be said in general. If $k>0$ is small enough, then the norm of $L\_k$ will be less than $1$ and hence $L\_k$ cannot be hypercyclic. For other values of $k$, the spectrum of $L\_K$ will be
$$
k(1-\sigma(L))
$$
and thus it may not be true that every component of $\sigma(L\_k)$ intersects the unit circle (and hence it will not be hypercyclic). My feeling (but I don't know for sure) that except for very special cases, $L\_k$ will not be hypercyclic (but I may be wrong).


