Left-Invertibility of Rank-One Perturbations

Susmita Das () and Jaydeb Sarkar ()
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Susmita Das: Statistics and Mathematics Unit, Indian Statistical Institute
Jaydeb Sarkar: Statistics and Mathematics Unit, Indian Statistical Institute

A chapter in Multivariable Operator Theory, 2023, pp 361-382 from Springer

Abstract: Abstract 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.

Keywords: Left-invertible operators; Rank-one perturbations; Shifts; Isometries; Diagonal operators; Reproducing kernel Hilbert spaces; 47A55; 47B37; 30H10; 47B32; 46B50; 47B07 (search for similar items in EconPapers)
Date: 2023
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-031-50535-5_13

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DOI: 10.1007/978-3-031-50535-5_13

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