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

Diagana, T.; McNeal, G. D.

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)].

Reviewer: Miroslav Zelený (Praha)

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

Keywords:

non-Archimedean Hilbert space; spectral analysis; diagonal operator; rank one operator

Citations:

Zbl 0930.47049