Regarding invertibility of a certain element in a Banach algebra

Question

Let $A$ be a complex unital Banach algebra. Let $a,b\in A$ such that $\sigma(a)\setminus \{0\}\subset \sigma(b)\setminus \{0\}$. Now let $\lambda\neq 0$ such that $\lambda\notin\sigma(b)$. Then $1+b(\lambda-b)^{-1}=\lambda(\lambda-b)^{-1}$ is invertible. Can anyone tell how does this imply $1+a(\lambda-b)^{-1}$ is invertible?

Answer 1

No such implication. Consider $a=1$, $b=\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} $, $\lambda=-2$.