In my research in the theory of Reproducing Kernel Hilbert Spaces I was concerned with this topic which came up but I could not find a reference on:

If $ \mathbb{H} $ is an RKHS and we denote the multiplier algebra by $ Mult(H) $ the algebra of multiplication operators on $ \mathbb{H} $ by multipliers of H, that is, those special functions f such that $ \forall h \in H : fh \in H $ and now we have $ Mult(H)' $ the commutant of $ Mult(H) $, that is all bounded operators on $ \mathbb{H} $ that commute with every multiplier operator on H, $ Mult(H)' = \{ T \in B(H) | \forall M \in Mult(H): MT=TM \}$

My question is can we deduce something (given that H is a RKHS) about the inclusion or intersection relation between the multiplier algebra and its commutant? It seems there should be something nice holding here, but would I be so lucky as to have inclusion or equality? It seems to good to be true but intuitively I can think of a few cases for this argument. Could someone please help me resolve this?