  For each isometry $V$ acting on some Hilbert space and a pair of vectors $f$
and $g$ in the same Hilbert space, we associate a nonnegative number $c(V;f,g)$
defined by \[ c(V; f,g) = (\|f\|^2 - \|V^*f\|^2) \|g\|^2 + |1 + \langle V^*f ,
g\rangle|^2. \] We prove that the rank-one perturbation $V + f \otimes g$ is
left-invertible if and only if \[ c(V;f,g) \neq 0. \] We also consider examples
of rank-one perturbations of isometries that are shift on some Hilbert space of
analytic functions. Here, shift refers to the operator of multiplication by the
coordinate function $z$. Finally, we examine $D + f \otimes g$, where $D$ is a
diagonal operator with nonzero diagonal entries and $f$ and $g$ are vectors
with nonzero Fourier coefficients. We prove that $D + f\otimes g$ is
left-invertible if and only if $D+f\otimes g$ is invertible.
