Hollow-core door shelves bracketed with 3 pieces of 1x1?
You should re-consider several of your ideas. 


The stripping on the wall is an excellent way to support shelves. However do use something like a 1x3 or 1x4 so that the material has some substance and strength. A 1x1 is just too flimsy, can bow between studs and after accepting one screw through it there is not much meat left to the piece. A wider strip allows for two screws at each stud.
I would purge the idea of using the hollow core doors for shelves. First off for a linens closet where you have multiple shelves the thickness of the doors ends up wasting storage space! Secondly after you cut hollow core doors for length and width you are left with open sides that only will show the internal web of cardboard. That will not standup over time. So unless you intend to re-glue side and end strips back into the edges of the cut off doors I would purge the idea of using them.


With the wall stripping I would think that a decent quality 3/4" plywood would make a much better shelf. If you have concerns of it bowing across the front you can add a 1x2 strip along under the front edge of the shelf that extends the full length. But honestly the plywood will be plenty strong for linens.