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- W4307511720 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 $$begin{aligned} c(V; f,g) = (Vert fVert ^2 - Vert V^*fVert ^2) Vert gVert ^2 + |1 + langle V^*f , grangle |^2. end{aligned}$$We prove that the rank-one perturbation (V + f otimes g) is left-invertible if and only if $$begin{aligned} c(V;f,g) ne 0. end{aligned}$$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), where D is a diagonal operator with nonzero diagonal entries and f and g are vectors with nonzero Fourier coefficients. We prove that (D + fotimes g) is left-invertible if and only if (D+fotimes g) is invertible." @default.
- W4307511720 created "2022-11-02" @default.
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- W4307511720 date "2022-10-28" @default.
- W4307511720 modified "2023-10-08" @default.
- W4307511720 title "Left-Invertibility of Rank-One Perturbations" @default.
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- W4307511720 doi "https://doi.org/10.1007/s11785-022-01295-8" @default.
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