Let $A$ be a commutative unital Banach algebra. We know that for any two elements of $A$ $\sigma(a+b)\subset \sigma(a)+\sigma(b)$ and $\sigma(ab)\subset \sigma(a)\sigma(b)$.
I know that if we consider $2 \times 2$ matrices, then we get examples of a non-commutative unital banach algebra such that $\sigma(a)+\sigma(b)\not\subset \sigma(a+b) $ and $\sigma(a)\sigma(b)\not\subset \sigma(ab)$.
I wonder if there are examples of $a$ and $b$ in a non-unital commutative $A$ such that $\sigma(a)+\sigma(b)\not\subset \sigma(a+b) $ and $\sigma(a)\sigma(b)\not\subset \sigma(ab)$. (I can't think of any at the moment)
Thank you in advance.
Best Answer
Let $A=\mathbb C^2\oplus C_0(\mathbb R)$. For the sum, you could do $$ a=\begin{bmatrix}1\\-1\end{bmatrix}\oplus0,\qquad b=\begin{bmatrix}-1\\1\end{bmatrix}\oplus0. $$ Then $$ \sigma(a)+\sigma(b)=\{-2,0,2\}\not\subset \{0\}=\sigma(a+b). $$ For the product, $$ a=\begin{bmatrix}1\\0\end{bmatrix}\oplus0,\qquad b=\begin{bmatrix}0\\1\end{bmatrix}\oplus0. $$ Then $$ \sigma(a)\sigma(b)=\{0,1\}\not\subset \{0\}=\sigma(ab). $$