English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We address the question of the analyticity of a rank one perturbation of ananalytic operator. If $\mathscr M_z$ is the bounded operator of multiplicationby $z$ 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 characterizedin terms of the membership to $\mathscr H_\kappa$ of the formal power seriesobtained by multiplying $f(z)$ by $\frac{1}{f(0)-z}.$ As an application, wediscuss the problem of the invariance of the left spectrum under rank oneperturbation. 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 $\inp{f}{g}.$ Ingeneral, 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 $T + f \otimes g.$

Analyticity, rank one perturbations and the invariance of the left spectrum