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- W4226504875 abstract "We address the question of the analyticity of a rank one perturbation of an analytic operator. If $mathscr M_z$ is the bounded operator of multiplication by $z$ on a functional Hilbert space $mathscr H_kappa$ and $f in mathscr H$ with $f(0)=0,$ then $mathscr M_z + f otimes 1$ is always analytic. If $f(0) neq 0,$ then the analyticity of $mathscr M_z + f otimes 1$ is characterized in terms of the membership to $mathscr H_kappa$ of the formal power series obtained by multiplying $f(z)$ by $frac{1}{f(0)-z}.$ As an application, we discuss the problem of the invariance of the left spectrum under rank one perturbation. In particular, we show that the left spectrum $sigma_l(T + f otimes g)$ of the rank one perturbation $T + f otimes g,$ $,g in ker(T^*),$ of a cyclic analytic left invertible bounded linear operator $T$ coincides with the left spectrum of $T$ except the point $inp{f}{g}.$ In general, the point $inp{f}{g}$ may or may not belong to $sigma_l(T + f otimes g).$ However, if it belongs to $sigma_l(T + f otimes g) backslash {0},$ then it is a simple eigenvalue of $T + f otimes g.$" @default.
- W4226504875 created "2022-05-05" @default.
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- W4226504875 date "2022-02-13" @default.
- W4226504875 modified "2023-09-24" @default.
- W4226504875 title "Analyticity, rank one perturbations and the invariance of the left spectrum" @default.
- W4226504875 doi "https://doi.org/10.48550/arxiv.2202.06433" @default.
- W4226504875 hasPublicationYear "2022" @default.
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