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An operator on some ultrametric Hilbert spaces. (English) Zbl 0930.47049
Let \(K\) be a complete valued field with valuation \(v\) and value group \(\Gamma\). Let \(\omega= (\omega_i)_{i\geq 0}\subset K\setminus\{0\}\) and let \(E_\omega\) be the Hilbert space associated with \(\omega\), i.e. the space of sequences \(x= (x_i)_{i\geq 0}\subset K\) such that \(\lim_{i\to\infty} x^2_i\omega_i= 0\). In the case where \(\omega_0= 1\), \(\lim_{i\to\infty} \omega^{-1}_i= 0\) and the sequence \((v(\omega_j))_{j\geq 0}\) is strictly decreasing in \(\Gamma\), one can define, as done by H. A. Keller and H. A. Ochsenius [Math. Slovaca 45, No. 4, 413-434 (1995; Zbl 0855.46049)], for a specific sequence \(\omega\) in a specific valued field \(K\), the operator \(A\) on \(E_\omega\) by setting on the canonical orthogonal base \((e_i)_{i\geq 0}\) of \(E_\omega\), \(Ae_i= e_i+ \sum_{j\neq i} \omega^{-1}_j e_j\). In the specific example studied by Keller and Ochsenius, the value group is \(\Gamma= \mathbb{Z}^{(\mathbb{N})}\) with the antilexicographic order. They showed that the operator \(A\) is topologically irreducible and \(\text{spec}(A)= \{1\}\).
In this paper, the author shows that in the rank one value case the situation is different. In fact, the operator \(A\) has eigenvalues and its eigenvalues are of the form \(\lambda= 1+\alpha\), where \(\alpha\) runs over the zeros of the function \(\phi(\alpha)= 1-\sum_{j\geq 0}{1\over 1+\alpha\omega_j}\). The zeros of \(\phi\) are located by classical \(p\)-adic analytic function theory. Moreover, it is shown that \(\text{spec}(A)= \{1\}\cup\{\text{eigenvalues of }A\}\) and \(1\) is the limit of the sequence of eigenvalues.

47S10 Operator theory over fields other than \(\mathbb{R}\), \(\mathbb{C}\) or the quaternions; non-Archimedean operator theory
47A75 Eigenvalue problems for linear operators
Zbl 0855.46049