I would say first that the question that is being asked here is a very deep one, and considerable care is warranted when trying to answer it. In that vein, we have to define the meaning of the terms we are using carefully. Category errors abound in this area. So, let's first be clear that the term "causality" refers to properties of physical processes, that take place in space and (crucially) time. Solutions of mathematical equations, on the other hand, are not physical processes so we need to make sure we are clear about what we mean by "causality properties" of solutions to (differential equations). This may sound like I'm needlessly splitting hairs, but here are some examples:
Example I: Projectile Motion:
Let's say we consider the motion of a projectile, which we could describe by a certain ordinary differential operator. Following my earlier comment, it is now important to note that we are going to solve an initial/boundary value problem, not an operator. As an analogy from linear algebra, the latter would be like saying "we are solving a coefficient matrix" rather than "we are solving a linear system of equations". Obviously, the former makes no sense. Now, going back to our projectile motion, a standard way to pose such a problem would be to provide initial values for the initial position and velocity of the projectile, and then solve the second-order ODE. Doing so seems to correspond closely to a causal process, where the progress of the projectile is causally determined from its prior state.
However, notice that we could pose a mathematically different problem, by providing the positions of the projectile at two points in time. Mathematically we would then be solving a boundary value problem, and the trajectory of the projectile would follow from a process that does not appear "causal" in the above sense at all, in that its details depend on boundary data at two end points. Yes, one might argue that this way to phrase the problem is somewhat "artificial", but the point is that we were dealing with the exact same differential operator, and may obtain the exact same solution.
Example II: Planetary Motion:
My next example is cribbed from the second Messenger Lecture Feyman gave at Cornell University in 1964, on "The Relation of Mathematics to Physics". In the lecture, Feynman discusses the problem of planetary motion. Using Newtonian Mechanics, we end up with an initial value problem to define the planet's trajectory. We could see the motion of the planet again as a process where the gravitational forces cause the planet to move in a certain way. At each step, the central force will "bend" the path of the planet a little so that eventually we end up with an ellipse. However, Feynman points out that we could use a radically different mathematical formulation of the exact same physics based on a variational principle. Such a formulation does not contain the concept of causality in an obvious way (I would argue, in any way at all). In this case the equations are different, but the solution is exactly the same.
And of course, we could instead move to the more advanced physical model of general relativity, in which case causality might again disappear when we speak of the planet simply following geodesics in a curved space-time.
Partial differential equations:
So let's talk about partial differential equations now. Just as for ODE's, and actually much more so, it is possible to formulate equivalent problems in very different ways. I could solve a problem described by an elliptic operator (which would typically be considered an "a-causal operator", I think) by providing boundary conditions that look like the ones for a hyperbolic operator, and my solution process could look like a "causal process". It is also worth pointing out that much more complex types of boundary conditions can and do appear e.g. in certain formulations of the fundamental equations of fluid flow (for example, a vorticity transport formulation of the Navier-Stokes equations for incompressible flow naturally requires boundary conditions in an integral form).
To summarize, I would say that I am quite doubtful that it is possible to classify differential operators as being "causal" or "having causal solutions". Based on what I said above one might be tempted to augment this notion by classifying, say, initial/boundary value problems (so, operator plus initial/boundary conditions) as having "causal solutions". However, I am very skeptical as to what such a classification would, or even could accomplish.
Finally, I have come to feel that the concept of causality in general may be problematic in principle, see John Norton's paper on "Causation as Folk Science". As a matter of fact, after all these years I am now less certain of my understanding of the concept of causality than I may have ever been. I want to emphasize that I offer the above more as food for thought and further discussion than a formal answer. The way I have come to know this forum, people may well complain that the above isn't really an answer at all. And they may be right...
