Examples for spectrum of the sum and product of two elements in a commutative Banach algebra.

Question

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 $\sigma(ab)\subset \sigma(a)\sigma(b)$. Can you give examples of $a$ and $b$ in $A$ such that $\sigma(a)+\sigma(b)\not\subset \sigma(a+b) $ and $\sigma(a)\sigma(b)\not\subset \sigma(ab) $ ?