Though this does not directly answer your question, here is a foundational paper that might help one derive results that might answer your question:
Marian Boykan Pour-El and Ian Richards: "Noncomputability in Analysis and Physics: A Complete Determination of the Class of Noncomputable Linear Operators", Advances in Mathematics 48, 44-74 (1983).
I quote the short first paragraph of this paper as it sets the tone for what follows:
"One would assume that a "reasonable" operator should map computable input data onto computable solutions. It is perhaps surprising that many of the standard operators of analysis and physics fail to do this. In this article, we shall determine precisely which linear operators do, and which do not, preserve computability."
I hope this paper helps.