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 $$\begin{aligned} c(V; f,g) = (\Vert f\Vert ^2 - \Vert V^*f\Vert ^2) \Vert g\Vert ^2 + |1 + \langle V^*f , g\rangle |^2. \end{aligned}$$ c ( V ; f , g ) = ( ‖ f ‖ 2 - ‖ V ∗ f ‖ 2 ) ‖ g ‖ 2 + | 1 + ⟨ V ∗ f , g ⟩ | 2 . We prove that the rank-one perturbation $$V + f \otimes g$$ V + f ⊗ g is left-invertible if and only if $$\begin{aligned} c(V;f,g) \ne 0. \end{aligned}$$ c ( V ; f , g ) ≠ 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 $$D + f \otimes g$$ D + f ⊗ g , where D is a diagonal operator with nonzero diagonal entries and f and g are vectors with nonzero Fourier coefficients. We prove that $$D + f\otimes g$$ D + f ⊗ g is left-invertible if and only if $$D+f\otimes g$$ D + f ⊗ g is invertible.