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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.

MSC:
 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