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 Mz is the bounded operator of multiplication by z on a functional Hilbert space Hκ and f ∈ H with f (0) = 0, then Mz + f ⊗ 1 is always analytic. If f (0) = 0, then the analyticity of Mz + f ⊗ 1 is characterized in terms of the membership to Hκ of the formal power series obtained by multiplying f (z) byAs 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 σ l (T + f ⊗ g) of the rank one perturbation T + f ⊗ g, g ∈ ker(T * ), of a cyclic analytic left invertible bounded linear operator T coincides with the left spectrum of T except the point f, g . In general, the point f, g may or may not belong to σ l (T + f ⊗ g). However, if it belongs to σ l (T + f ⊗ g)\{0}, then it is a simple eigenvalue of T + f ⊗ g.