I was wondering about the general definition of a quasicompact operator. There seem to be two main ones floating around in the literature, and I am not sure they are equivalent. The first, and seemingly most prevalent one is that an operator $T:B \to B$ is quasi-compact, if there exists a natural number $n$, and a compact operator $K$, so that $$ \|T^n-K\|<1 $$ The second definition says that $T$ is quasi-compact, if there exists two closed, invariant subspaces $F, H$, so that $$ B=F\oplus H $$ with $dim(F) <\infty$, so that the spectral radius of $T$ restricted to $H$ is strictly less than the spectral radius of $T$, $r(T)$, and any eigenvalue of $T$ restricted to $F$ has magnitude $r(T)$.
My main query is that, whilst these definitions seem to be equivalent in the special case when $r(T) = 1$, the second definition seems to imply that the class of quasi-compact operators is stable under scalar multiplication, whilst the first does not. I believe I came up with an example which I will put at the end of the question. A seemingly simple fix would be to require that $\|T^n-K\| < r(T)^n$, but I haven't seen it used anywhere. I guess since these operators mainly show up in ergodic theory, having unit norm is very natural, and so perhaps this is how the convention was born. Nevertheless, having read quite a few papers now which use definition 1 in the form I stated, I was still wondering if perhaps I had perhaps missed something.
$\textbf{Example:}$
Let $B$ be a Hilbert space with orthonormal basis $e_i$. Define now an operator $T:B \to B$ by $Te_i=e_i$ for $i >1$, and $Te_1=2e_1$. This operator is quasi-compact in the sense of definition 2 (take $F=span\{e_1\}$, $H=\overline{span\{e_i:i\geq 2\}}$), but not the sense of definition 1. However, $\frac{T}{\|T\|}$ is again quasi-compact in the sense of definition 1. To see $T$ is not quasicompact in the sense of definition 1, note that $T^n$ still agrees with $T$ on any $e_i$ with $i \geq 2$. Thus, if $K$ is compact and $\|T^n-K\|<1$, there must exists a constant $a>0$ so that for all $i \geq 2$, it holds $$ |1-\langle e_i,Ke_i\rangle|\leq \|T^ne_i-Ke_i\|\leq 1-a $$ so that $a \leq |\langle e_i,Ke_i\rangle|$. By the compactness assumption, since the sequence $\{e_i\}$ converges weakly to zero, $Ke_i$ must converge to zero in norm. But then $\|Ke_i\|\geq a$ for any $i \geq 2$, so this cannot be.