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

Abstract

We address the question of the analyticity of a rank one perturbation of an analytic operator. If ${\mathcal{M}}_z$ is the bounded operator of multiplication by $z$ on a functional Hilbert space ${\mathcal{H}}_{\kappa }$ and $f\in \mathcal{H}$ with $f(0)=0,$ then ${\mathcal{M}}_z+f\otimes 1$ is always analytic. If $f(0)\ne 0,$ then the analyticity of ${\mathcal{M}}_z+f\otimes 1$ is characterized in terms of the membership to ${\mathcal{H}}_{\kappa }$ 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 $T+f\otimes g,$ $g\in \ker (T^{\ast }),$ 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 $T+f\otimes g.$