The results presented in this paper are motivated by some of the results obtained by B.~Beauzamy in \cite[Chap. III]{bb1} for a single operator on an infinite-dimensional complex Hilbert space that imply existence of a dense set of vectors with orbits tending strongly to infinity. For the case of invertible operator $T$, one of B.~Beauzamy's results implies that the space actually contains a dense set of vectors for which both the orbits under $T$ and its inverse tend strongly to infinity. We are going to show that this is also true for any suitable pair of operators.