The paper is concerned with the spectral analysis for the class of linear operators $A=D_{\lambda }+X\otimes Y$ in non-archimedean Hilbert space, where $D_{\lambda }$ is a diagonal operator and $X\otimes Y$ is a rank one operator. The results of this paper turn out to be a generalization of those results obtained by Diarra.

Let ${\epsilon }_{\omega }\left(I\right)$ denote the space of all ω-ultradifferentiable functions of Roumieu type on an open interval I in ℝ. In the special case ω(t) = t we get the real-analytic functions on I. For $\mu \in {\epsilon }_{\omega }{\left(I\right)}^{\text{'}}$ with $supp\left(\mu \right)=0$ one can define the convolution operator $T_{\mu }:{\epsilon }_{\omega }\left(I\right)\to {\epsilon }_{\omega }\left(I\right)$, $T_{\mu }\left(f\right)\left(x\right):=⟨\mu ,f(x-·)⟩$. We give a characterization of the surjectivity of $T_{\mu }$ for quasianalytic classes ${\epsilon }_{\omega }\left(I\right)$, where I = ℝ or I is an open, bounded interval in ℝ. This characterization is given in terms of the distribution of zeros of the Fourier Laplace transform $\widehat{\mu }$ of μ.