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Spectral analysis for finite rank perturbations of diagonal operators in non-Archimedean Hilbert space. (English) Zbl 1317.47069

Let \(\mathbb{K}\) be a non-Archimedean nontrivially valued field that is complete under the ultrametric induced by its valuation \(\left| \, . \, \right|\). Fix a sequence \(w = (w_j)_{j }\) of nonzero elements of \(\mathbb{K}\). Define the space \(E_w:= \{ u= (u_j)_{j} \in \mathbb{K}^{\mathbb{N}} : \lim_j \left| u_j \right| \left| w_j \right|^{1/2} =0 \}\). It is well known that the space \(E_w\) equipped with the norm defined for each \(u = (u_j)_{j } \in E_w\) by \(\| u \|:= \sup_{j} \left| u_j \right| \left| w_j \right|^{1/2} \) is a non-Archimedean Banach space over \(\mathbb{K}\). Similarly, an inner product is also defined on \(E_w\) for all \(u = (u_j)_{j}, v = (v_j)_{j } \in E_w\) by \(\langle u,v\rangle = \sum_j w_j u_j v_j\). In the literature, the space \(E_w\) endowed with the above mentioned norm and inner product is called a non-Archimedean Hilbert space (in contrast with the classical case, the norm on \(E_w\) does not stem from the inner product).
The main purpose of this paper is to study the spectral analysis for the class of finite-rank perturbations of diagonal operators on \(E_w\), i.e., operators \(T = D + F\), where \(D\) is a diagonal operator with \(D e_k = \lambda_k\) (here, \(\lambda_1, \lambda_2, \dots\) is a bounded sequence in \(\mathbb{K}\) and \(e_1, e_2, \dots\) are the unit vectors of \(E_w\)) and \(F = u^1 \otimes v^1 + u^2 \otimes v^2 + \dots+ u^m \otimes v^m\) is an operator of rank at most \(m\) with \(u^k, v^k \in E_w \setminus \{ 0 \}\) for \(k = 1, \dots, m\). The Fredholm operator theory existing in the non-Archimedean setting is a useful tool to get that purpose.
Firstly, the authors prove that the spectrum \(\sigma(T)\) of \(T\) is given by \(\sigma(T) = \{ \lambda \in \rho(D) : \text{ det} \, M(\lambda) =0 \} \cup \sigma_e(D)\). Here, \(\rho(D)\) is the resolvent of \(D\) (i.e., \(\{ \lambda \in \mathbb{K} : \lambda I - D\) is a bijection and its inverse is continuous\(\}\)), \(\sigma_e(D)\) is the essential spectrum of \(D\) (i.e., \(\{ \lambda \in \mathbb{K} : \lambda I - D\) is not a Fredholm operator of index \(0\}\)), and \(M(\lambda) = (c_{ij}(\lambda))\) with \(c_{ij}(\lambda) = \delta_{ij} + \langle C_{\lambda} u^j, v^i\rangle\) (\(i,j = 1, \dots, m\)) and \(C_{\lambda}:= (D - \lambda I)^{-1}\).
Then they compute \(\sigma_e(D)\) from which the main theorem of the paper, Theorem 6.1, is obtained. This states that \(\sigma(T) = \{ \lambda \in \rho(D) : \operatorname{det} M(\lambda) =0 \} \cup [ (\overline{\Lambda} \setminus \Lambda^{*}) \cup ( \Lambda^{*} \cap \Lambda')]\), where \(\Lambda = \{ \lambda_j : j \in \mathbb{N} \}\), \(\overline{\Lambda}\) is the closure of \(\Lambda\) in \(\mathbb{K}\), \(\Lambda^{*}\) is formed by the \(\alpha \in \Lambda\) for which the set \(\{ j \in \mathbb{N}: \lambda_j = \alpha \}\) is finite, and \(\Lambda'\) is formed by the \(\alpha \in \Lambda\) that are accumulation points of \(\Lambda\).
Finally, the authors give a few examples at the end of the paper to illustrate their results.
The fact that in the non-Archimedean context the compact operators are limits of sequences of finite rank operators, leads to ask whether Theorem 6.1 of the paper admits an extension to the more general case of compact perturbations of diagonal operators. This would be an interesting question to study in a future research work.

MSC:

47S10 Operator theory over fields other than \(\mathbb{R}\), \(\mathbb{C}\) or the quaternions; non-Archimedean operator theory
12J25 Non-Archimedean valued fields
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References:

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