Jul 31, 2012 Thread starter #1 D dray Member Feb 5, 2012 37 Let $X$ be a *-algebra with identity $e$, and let $e\in{X}$, $\lambda\in\mathbb{C}$. Can somebody show me how $\sigma(\lambda{e}-x)=\lambda-\sigma(x)$, where $\sigma(x)$ is the spectrum of an element. Thanks in advance.

Let $X$ be a *-algebra with identity $e$, and let $e\in{X}$, $\lambda\in\mathbb{C}$. Can somebody show me how $\sigma(\lambda{e}-x)=\lambda-\sigma(x)$, where $\sigma(x)$ is the spectrum of an element. Thanks in advance.

Jul 31, 2012 #2 G girdav Member Feb 1, 2012 96 $v\in\sigma(\lambda e-x)$ if and only if $(\lambda-v)e-x$ is invertible, that is if and only if $\lambda-v\in\sigma(x)$, which gives the result.

$v\in\sigma(\lambda e-x)$ if and only if $(\lambda-v)e-x$ is invertible, that is if and only if $\lambda-v\in\sigma(x)$, which gives the result.