  In this paper we study in a Hilbert space a homogeneous linear second order
difference equation with nonconstant and noncommuting operator coefficients. We
build its exact resolutive formula consisting in the explicit non-iterative
expression of a generic term of the unknown sequence of vectors of the Hilbert
space. Some non-trivial applications are reported with the aim of showing the
usefulness and the broad applicability of our result.
