## Spectral analysis for rank one perturbations of diagonal operators in non-Archimedean Hilbert space.(English)Zbl 1212.47126

Commentat. Math. Univ. Carol. 50, No. 3, 385-400 (2009); corrigendum ibid. 50, No. 4, 637-638 (2009).
Let $$(\mathbb K, | \cdot | )$$ be a nontrivial complete non-Archimedean valued field. Let $$\omega = (\omega _j)_{j\in \mathbb N}$$ be a fixed sequence of nonzero elements of $$\mathbb K$$. Under some additional technical assumptions, the authors compute the spectrum of each bounded linear operator $$A$$ on the non-Archimedean Hilbert space $$c_0(\omega ,\mathbb N,\mathbb K)$$ of the form $$A = D_{\lambda } + X \otimes Y$$, where $$D_{\lambda }$$ is a diagonal operator and $$X \otimes Y$$ is a rank one operator with $$X,Y \in c_0(\omega ,\mathbb N,\mathbb K)$$. The paper provides a generalization of results obtained by B. Diarra [J. Anal. 6, 55–74 (1998; Zbl 0930.47049)].

### MSC:

 47S10 Operator theory over fields other than $$\mathbb{R}$$, $$\mathbb{C}$$ or the quaternions; non-Archimedean operator theory 46S10 Functional analysis over fields other than $$\mathbb{R}$$ or $$\mathbb{C}$$ or the quaternions; non-Archimedean functional analysis

Zbl 0930.47049
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