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

TL;DR

This paper investigates the analyticity of rank one perturbations of analytic operators and explores how such perturbations affect the invariance of the left spectrum, providing conditions for spectrum changes and eigenvalue properties.

Contribution

It characterizes the analyticity of rank one perturbations of multiplication operators and analyzes the invariance of the left spectrum under these perturbations.

Findings

Rank one perturbations preserve analyticity under certain conditions.

The left spectrum of a perturbed operator mostly remains invariant, except possibly at a specific point.

If the point belongs to the spectrum, it is a simple eigenvalue.

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 $T + f \otimes g.$