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

Abstract

We discuss 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 $f(0) \ne 0,$ 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 $\langle {f},\,{g} \rangle .$ In general, the point $\langle {f},\,{g} \rangle$ 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 $T + f \otimes g$.