suppose that $A$ is a Banach algebra with identity, $a, b \in A$:
a) are these correct?
$ \sigma(a+b) \subseteq \sigma(a) +\sigma(b) $
$ \sigma(ab)\subseteq \sigma(a)\sigma(b) $
can you show me that abelian condition $A$ is necessary?
b)if $ab=ba$,
$ \sigma(a+b) \subseteq \sigma(a) +\sigma(b) $
$ \sigma(ab)\subseteq \sigma(a)\sigma(b) $