Let $A$ be a bounded linear operator from a Banach space $M$ to itself. Suppose that $\rho(A)<1$ where $\rho(A)$ is the spectral radius of $A$. For any $\varepsilon>0$, does there exist an open neighborhood $U$ of the origin such that $U\subset B(0,\varepsilon)$ and $A(U)\subset U$, where $B(0,\varepsilon)=\{x\in M | \|x\|<\varepsilon\}$?
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