the banach algebra

Question

let $A$ be a Banach algebra with identity,$\lambda \in \mathbb{C}$, is it right?

$\sigma(a + \lambda 1 ) = \sigma(a) +\lambda$$\qquad$$(a \in A)$

$\sigma (a) = \{ \lambda \in \mathbb{C} : \lambda 1 - a \qquad \text{is not invertible} \} $$\qquad$$(a \in A)$

Answer 1

Hint. Let $a' := a + \lambda 1$. Then $$ \sigma(a+\lambda 1) = \sigma(a') = \{\mu \in \mathbf C: \mu 1 - a' \text{ is not invertible}\} $$ Now use $$ \mu 1 - a' = \mu 1 - (a + \lambda 1) = (\mu - \lambda)1 - a $$