Left-Invertibility of Rank-One Perturbations

Article Type

Research Article

Publication Title

Complex Analysis and Operator Theory


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+⟨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 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⊗ 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⊗ g is left-invertible if and only if D+ f⊗ g is invertible.



Publication Date



Open Access, Green

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