It is written in Bratteli-Robinson that some simple transformations yield the relations:
$$\sigma(a+A)=a+\sigma(A)$$
$$\sigma(A^n)\subseteq\sigma(A)^n$$
The latter one is deduced by the transformation:
$$\lambda^n-A^n=\ldots=(\lambda-A)(\ldots)$$
Then I guess the result follows by:
$$C\text{ invertible }\iff A,B\text{ invertible }\qquad\text{for }C=AB$$
How do I prove this corollary?
Is there a similar trick for the former relation?
...oh, actually the corollary above is wrong:
Neither the right shift nor the left shift operator are invertible but their composite: $\mathbb{1}=LR$
Since the corollary only holds in the finite dimensional case (if at all) then the argumentation followed in their book is not right.
Suppose $C=AB$ and $AB=BA$. Then what can you say?