  In the present paper, we consider a Cauchy problem for a linear second order
in time abstract differential equation with pure delay. In the absence of
delay, this problem, known as the harmonic oscillator, has a two-dimensional
eigenspace so that the solution of the homogeneous problem can be written as a
linear combination of these two eigenfunctions. As opposed to that, in the
presence even of a small delay, the spectrum is infinite and a finite sum
representation is not possible. Using a special function referred to as the
delay exponential function, we give an explicit solution representation for the
Cauchy problem associated with the linear oscillator with pure delay. In
contrast to earlier works, no positivity conditions are imposed.
