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 $L_k$?
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$\begingroup$ Do you mean $L_k:=k(I_B-L)$? $\endgroup$– Ruben A. Martinez-AvendanoCommented May 19, 2020 at 15:27
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$\begingroup$ Yes, thanks for pointing this out $\endgroup$– ABIMCommented May 19, 2020 at 22:38
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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).
answered May 20, 2020 at 23:31
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1$\begingroup$ Thanks Ruben, I had a similar intuition. $\endgroup$– ABIMCommented May 21, 2020 at 7:37