Addition and multiplication of Spectrums

Question

I know that if $A$ is a unital abelian Banach algebra, then we have $\sigma(x+y)\subset \sigma(x)+ \sigma(y)$ and $\sigma(xy)\subset \sigma(x)+\sigma(y)$.

So we have the notation $\sigma(x)+\sigma(y)$ and $\sigma(x) \sigma(y)$. So I wonder how the addition and multiplication are defined.

For example, if we consider the set of $2\times 2$ matrices.

Let $a=\pmatrix{-1&0\\0&1}$, $b=\pmatrix{1&1\\0&0}$.

Then $\sigma(a)= \{ 1, -1\}$ , $\sigma(b)= \{1, 0 \}$

What are $\sigma(x)+\sigma(y)$ and $\sigma(x) \sigma(y)$ in this case?

Thank you!

Answer 1

For any sets $A$ and $B$, $A + B = \{a+b: a \in A,\; b \in B\}$ and similarly $AB = \{ab: a \in A,\; b \in B\}$.

Note that in your example $a$ and $b$ don't commute, so they are not relevant to the theorem.