This is 'the' kitchen light? Task lighting for work zones is the first of three layers used in kitchen lighting plans. The minimum norm is 450 lumens at the counter. The next layer is ambient lighting for the space as a whole. Finally a pendant is an example of decorative lighting. Under cabinet lighting is one type of task lighting.

Thank you for these suggestions and for the education regarding kitchen lighting plans. I love the fixture suggestions but I'm wondering if the width of the light should be shorter than the countertop width? The fixture width was one of the reasons that I hadn't previously considered a linear fixture but they are quite beautiful and would provide better light. If I select a linear fixture what is the maximum width that I should consider with a 42 in wide island countertop?

IMO both island lighting and DR lighting should light the surface below them not the ceiling. . I need to see your spaces for sure lighting is not chosen without context . If you have no other lighting in the kitchen then you need more lighting than on center fixture so I need pics poste here in a comment DO NOT start another post .

So your question is regarding the length of the island, correct? You said in your original post that the island is 36" x 42" so the dimension that you should consider is the length of the linear fixture. I think that the width of most of the linear fixtures would work just fine with a 36" wide island, since none of them are particularly wide. But with a 42" island you probably should use about a 30-33" long linear fixture. It depends on where the island is and the configuration of the fixture. You might want to post photos and a floor plan to get better advice.