Left-invertibility of rank-one perturbations

DOI10.1007/S11785-022-01295-8zbMATH Open1517.47022arXiv2201.10535MaRDI QIDQ6389169

Susmita Das, Jaydeb Sarkar

Publication date: 25 January 2022

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 [ c(V; f,g) = (|f|^2 - |V^*f|^2) |g|^2 + |1 + langle V^*f , g angle|^2. ] We prove that the rank-one perturbation $V+fotimesg$ is left-invertible if and only if [ c(V;f,g) eq 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 $D+fotimesg$, where $D$ is a diagonal operator with nonzero diagonal entries and $f$ and $g$ are vectors with nonzero Fourier coefficients. We prove that $D+fotimesg$ is left-invertible if and only if $D+fotimesg$ is invertible.

Mathematics Subject Classification ID

Linear operators defined by compactness properties (47B07) Perturbation theory of linear operators (47A55) Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.) (47B37) Compactness in Banach (or normed) spaces (46B50) Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces) (47B32) Hardy spaces (30H10)

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