Let $T$ be bounded linear operator in a Banach space $E$ such that a sequence $x, Tx, T^{2}x, T^{3}x, \dots$ is bounded for all $x \in E$. Show that if $\lambda \in \sigma(T)$ (spectrum of $T$), then $|\lambda| \leq 1$
Show that if $\lambda \in \sigma(T)$ (spectrum of $T$), then $|\lambda| \leq 1$
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functional-analysis
banach-spaces
spectral-theory
1 Answers
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Consider the family of operators $T^n$ where $n$ is an integer. The fact that $x,...T^n(x)$ is bounded implies there exists $A$ such that $\|T^n\| The spectral radius formula says that $r(T)=lim_n\|T^n\|^{1/n}\leq lim_nA^{1/n}=1$. This implies that for every $\lambda$ in the spectrum of $T$, $|\lambda|\leq 1$.