Let a,b be 2 elements in a Banach Algebra.Let Spec(x) denote the spectrum of an element x. If a,b commute with each other, then by Gelfand Transformation, we have Spec(a+b) is a subset of Spec(a)+Spec(b)& Spec(ab) is a subset of Spec(a)Spec(b) (Where the definition of addition and multipication of two sets is obvious).
My Questions, if we drop the the condition "a,b commute", then to what extent is the above relations about spectra still hold?