Left-Invertibility of Rank-One Perturbations

Abstract

For each isometry VVV 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+⟨V∗f, g⟩∣2. We prove that the rank‑one perturbation V+f⊗g is left‑invertible if and only if c(V;f,g)≠0. We prove that the rank-one perturbation of V+f⊗g isometries that are shifts on some Hilbert spaces of analytic functions here, shift to the operator of multiplication by the coordinate function z. Finally, we examine D+f⊗g where D is a diagonal operator with nonzero diagonal entries and f, and g with nonzero Fourier coefficients. We prove that D+f⊗g is left‑invertible if and only if D+f⊗g is invertible.