Non-Archimedean antiderivations and calculus of operators with local spectra (Q2795883)

This paper deals with spectral theory of operators on non-Archimedean Banach spaces over local fields that are quadratic extensions, \(\mathbb{K}(\alpha)\), of non-Archimedean local fields \(\mathbb{K}\).NEWLINENEWLINESections 2 and 3 are devoted to bounded operators. In Section 2, a spectral calculus for operators with local spectrum is discussed. Since the spectra of such operators are contained in non-Archimedean oriented boundaries of compact analytic oriented manifolds modeled on \(\mathbb{K}(\alpha)\), the author uses the line antiderivation operators on those boundaries, previously considered in [the author, Int. J. Math. Math. Sci. 2005, No. 2, 263--309 (2005; Zbl 1083.32020)]. Among other things, it is proved that the spectrum \(\sigma(T)\) of such an operator \(T\) is non-empty and that \(\sup \left| \sigma (T) \right| =\) the spectral radius of \(T:= \lim_m \| T^m \|^{1/m} \leq \| T \|\). In Section 3, the author studies the behavior of the spectrum under small perturbations. Finally, in Section 4, the case of unbounded operators is treated.NEWLINENEWLINEI think that for future research, it would be interesting to investigative whether the results of this paper are valid when the local compactness assumption on the non-Archimedean ground field is dropped. When working over local fields, we have several ``comforts'' related to spectral theory of operators, for instance:{\parindent=0.6cm \begin{itemize}\item[--]the compact sets of these fields are the closed and bounded ones; \item [--]every non-Archimedean Banach space over these fields is isomorphic to \(c_0(I,s)\) for certain \(I,s\). NEWLINENEWLINE\end{itemize}}When the ground field is not locally compact (e.g., the field \(\mathbb{C}_p\), the completion of the algebraic closure of the field \(\mathbb{Q}_p\) of \(p\)-adic numbers), the above two facts are false. Therefore, a non-local version of the present paper would appear to be very intriguing.