Common examples of linear operators are
- Matrices in finite dimensional spaces.
- Differentiation and definite and indefinite integration in appropriately regular functional spaces.
I remember that the second example was quite unexpected for me when I first learned of it in my non-mathematics undegraduate time. Some more examples that come up are boundary and coboundary operators in co-/homology theories, exterior derivative and boundary operator for the Stokes' theorem. Maybe some more are vector- and scalar product when one vector is fixed. But I cannot think of something more exotic at the moment.
What are some of your favorite uncommon examples of linear operators? Since linear operators need vector spaces to act on, I ask also for examples of uncommon vector spaces, but I'm primarily interested in linear operators.