@DCM the background of my research consists of operators that have an $AC[a,b]$ functional calculus and their integral representations. Trying to generalise the theory to compact subsets of $\mathbb{C}$ with an appropriate class of functions is why I was wondering if this object has been studied.

Sorry, I have corrected the norm. It should be submultiplicative now. As for the operation, it is pointwise multiplication. I'll edit that in too.