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

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 \(T + f \otimes g\).