Left-invertibility of rank-one perturbations

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 $$c(V;f,g)=(\mathrm{\parallel }f{\mathrm{\parallel }}^2-\mathrm{\parallel }{V}^{\ast }f{\mathrm{\parallel }}^2)\mathrm{\parallel }g{\mathrm{\parallel }}^2+\mathrm{\mid }1+⟨V^{\ast }f,g⟩{\mathrm{\mid }}^2\mathrm{.}$$ We prove that the rank-one perturbation $V+f\otimes g$ is left-invertible if and only if $$c(V;f,g)\mathrm{\ne }0.$$ 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.