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I can't solve the following:

i) Let $T:l^2 \rightarrow l^2$ , $Tx=\{ (Tx)_n\}_{n=1}^{\infty}$ given by $$(Tx)_n = \dfrac{1}{2}x_{n-1} + \dfrac{1}{2}x_n.$$ Find $\sigma(T)$.

ii) Let $S : l^2 \rightarrow l^2$ defined by $l(x_1, x_2, x_3, \dots ) = (x_2, x_3, \dots )$ Find $\sigma(S)$.

iii) Let $X$ be a normed vector space and $T\colon X \rightarrow X$ linear operator such that $T^{-1}$ exists. Show that $$ \sigma(T^{-1}) = \{ \lambda^{-1} : \lambda \in \sigma (T) \}.$$ Thanks in advance.

Davide Giraudo
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user73577
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1 Answers1

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For paragraph ii) see these notes. In fact $\sigma(S)=\{\lambda\in\mathbb{C}:|\lambda|\leq 1\}=\operatorname{Ball}_{\mathbb{C}}(0,1)$.

For paragraph iii) see theorem $1.3$ in this answer

For paragraph i) note that $T=0.5(1_{\ell_2}+S)$, then $$ \lambda\in\sigma(S)\Longleftrightarrow 0.5(1+\lambda)\in\sigma(T) $$ hence $\sigma(T)=\operatorname{Ball}_{\mathbb{C}}(1,0.5)$

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Norbert
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