I am looking for a way to prove that for two commuting, linear and bounded operators $A,B$ acting on a Banach space $X$ $$\sigma(A-B)\subset\sigma(A)-\sigma(B)$$
I have already found a proof that "simply" applies results from Gelfand theory. This however requires extensive knowledge on Banach Algebras.
That is why I would be happy to get some different ideas for ways of proving the statement above.