Cramer-type explicit determinantal solution formulas for the Fredholm equation $x - a\,u(x) = y$ with completely continuous endomorphisms of $p$-adic Banach spaces (Serre, 1962)

In his 1962 IHÉS paper, Serre established Fredholm–Riesz theory for completely continuous endomorphisms of non-archimedean Banach spaces: the Fredholm determinant $H(t) = \det(1 - tu)$ is an entire function, $1 - au$ is invertible if and only if $H(a) \neq 0$, and the solution of $x - a\,u(x) = y$ is given abstractly by $x = P(a,u)y/H(a)$ in terms of the Fredholm resolvent $P(t,u)$. He then conjectured (Section 7, Remark 2) that the explicit Cramer-type formulas of classical determinant theory in Banach spaces (the Leżański–Ruston–Sikorski 'Fredholm formulae', which express the solution through Fredholm minors) remain valid in the $p$-adic setting, while noting that Grothendieck's archimedean presentation cannot be transplanted because its factorial denominators have unbounded $p$-adic norm. Gruson (1966) subsequently extended the framework to arbitrary ultrametric Banach spaces and developed the completed exterior powers, but the minor-based solution formulas have not been developed, so the conjecture remains open. The problem is to construct, and prove convergent, explicit determinantal (Cramer/minor-type) formulas for the solution of $x - a\,u(x) = y$ in the non-archimedean completely continuous setting, or to prove that no such formulas exist.

Background

Let $K$ be a complete field with a non-archimedean absolute value (a '$p$-adic' or ultrametric valued field; the $p$-adic numbers $\mathbb{Q}_p$ and their finite extensions are the model cases), with valuation ring $A$, maximal ideal $\mathfrak{m}$, and value group $G = |K^\times| \subset \mathbb{R}_{>0}$. A Banach space over $K$ is a complete normed $K$-vector space whose norm satisfies the ultrametric inequality $|x + y| \le \sup(|x|, |y|)$; the unit ball $E_0$ is then an $A$-module. An orthonormal base of $E$ is a family $(e_i)_{i \in I}$ such that every $x \in E$ is uniquely $x = \sum x_i e_i$ with $|x| = \sup |x_i|$; the model example is the space $c(I)$ of families $(x_i)_{i \in I}$ tending to $0$ along the cofinite filter, with the sup norm. An endomorphism $u$ of $E$ is completely continuous if it is a norm-limit of finite-rank operators (over locally compact $K$ this coincides with mapping bounded sets to relatively compact sets). Serre proved: (i) when the valuation is discrete, every Banach space satisfying condition (N) (norms of elements lie in the closure of $G$) is isometric to some $c(I)$; (ii) for $E \cong c(I)$ and $u$ completely continuous, the Fredholm determinant $H(t) = \det(1 - tu)$, built by reducing the matrix of $u$ modulo the ideals $\mathfrak{m}^k$ and passing to the limit, is an entire function whose power-series coefficients are $(-1)^m \operatorname{Tr}(\wedge^m u)$ via completed exterior powers; (iii) the Fredholm resolvent $P(t,u) = \sum_{m \ge 0} v_m t^m$ is an entire $\mathcal{L}(E)$-valued function satisfying $(1 - tu) P(t,u) = H(t)$; (iv) $1 - au$ is invertible if and only if $H(a) \neq 0$, and then $(1 - au)^{-1} = P(a,u)/H(a)$ (Prop. 11); (v) a Riesz decomposition $E = N(a) \oplus F(a)$, where $N(a)$ is a finite-dimensional closed stable subspace on which $1 - au$ is nilpotent (of dimension equal to the order of vanishing of $H$ at $a$) and $F(a)$ is a closed stable complement on which $1 - au$ is invertible, and the Fredholm alternative hold. Thus the equation $x - a\,u(x) = y$ is completely solved in the abstract. On the archimedean side, a parallel 'determinant theory in Banach spaces' was developed by Leżański, Ruston, and Sikorski: starting from a chosen vector and functional, one builds determinant series (first and second characteristic functions, Fredholm minors) that solve $x - a\,u(x) = y$ by explicit Cramer-type formulas — ratios of convergent determinant series, mimicking finite-dimensional Cramer's rule. Serre's remark is that these explicit formulas 'very probably' persist in the $p$-adic setting; the identified obstruction is that the archimedean constructions normalize antisymmetrizations by factorials $n!$ (Grothendieck's 1956 presentation being the extreme case), and $|1/n!| \to \infty$ $p$-adically, so the classical series cannot be transplanted verbatim. Gruson (1966) subsequently rebuilt the whole theory for arbitrary ultrametric Banach spaces (no orthonormal base, no discrete valuation, using topological tensor products and completed exterior powers as quotients rather than averaged subspaces) and extended it to locally convex spaces, but his solution formula is still the recursive resolvent $P(a,u)y/H(a)$; the minor-based formulas were not developed.

Problem Statement

Let $K$ be a complete non-archimedean valued field, $E$ a Banach space over $K$, and $u : E \to E$ a completely continuous endomorphism (a norm-limit of finite-rank operators; the original setting is Serre's $E \cong c(I)$, and Gruson's framework allows arbitrary ultrametric Banach $E$). Let $H(t) = \det(1 - tu)$ be the Fredholm determinant and $P(t,u)$ the Fredholm resolvent; it is known that for $a \in K$ with $H(a) \neq 0$, the unique solution of the Fredholm equation $x - a\,u(x) = y$ is $x = P(a,u)y/H(a)$. Problem (Serre, 1962, Section 7, Remarque 2, stated as a conjecture: 'it is very probable that these formulas remain valid in the case considered here'): construct, or refute the possibility of, explicit Cramer-type determinantal formulas for this solution in the non-archimedean setting, i.e. a non-archimedean analogue of the Leżański–Ruston–Sikorski Fredholm formulae. The conjecture is that, for each $y \in E$, there exist non-archimedean analogues of the Fredholm minors (a second characteristic function attached to $y$ and $u$) as explicit determinant-type quantities such that the solution is expressed as a ratio of convergent determinant series, reducing to classical Cramer formulas when $u$ has finite rank; the numerator is itself obtained by a minor/determinant construction attached to $y$ (the second characteristic function), so that the known resolvent expression $P(a,u)y/H(a)$ is not by itself a formula of this kind. It is further conjectured that every series involved converges in the $p$-adic topology, with explicit norm bounds, in Serre's original setting ($E \cong c(I)$ over an arbitrary complete non-archimedean field $K$) and, ideally, in Gruson's general setting, and that no factorial denominators (or other normalizations of unbounded $p$-adic norm) are required, addressing Serre's observation that the archimedean constructions of Sikorski and of Grothendieck cannot be transplanted because $|1/n!| \to \infty$ $p$-adically. For the refutation branch, the class to be excluded is that of all expressions of the solution as a ratio of convergent local determinant/minor series built from completed exterior or tensor powers of $u$ by a limiting minor construction (the non-archimedean analogues of the classical first and second characteristic functions): the conjecture is refuted precisely when no expression in this class yields such a formula. The divergence of one conveniently chosen subclass of candidate series is not a refutation of the conjecture: Serre himself observed that the verbatim archimedean constructions with factorial denominators diverge $p$-adically, and this known obstruction is already part of the background, not an answer to the conjecture. The question is to be answered at least in Serre's original setting; Gruson's general setting contains it, so an answer there covers the original one.

References

1. Jean-Pierre Serre, Endomorphismes complètement continus des espaces de Banach p-adiques, Publications Mathématiques de l'IHÉS 12 (1962), 69-85, doi:10.1007/BF02684276, https://numdam.org/item/PMIHES_1962__12__69_0/ (primary source; the conjecture is Section 7, Remarque 2, p. 82; the abstract solution formula is Prop. 11 with the resolvent of Section 6)
2. L. Gruson, Théorie de Fredholm p-adique, Bulletin de la Société Mathématique de France 94 (1966), 67-95, doi:10.24033/bsmf.1635, https://numdam.org/item/BSMF_1966__94__67_0/ (generalizes Serre's framework to all ultrametric Banach spaces over any complete non-archimedean field, develops completed exterior powers and a Grothendieck-style presentation, proves the Riesz decomposition and locally convex extensions)
3. Roman Sikorski, The determinant theory in Banach spaces, Colloquium Mathematicum 8 (1961), no. 2, 141-198, doi:10.4064/CM-8-2-141-198, https://doi.org/10.4064/CM-8-2-141-198 (the classical exposition of the Cramer-type Fredholm formulas that Serre's conjecture refers to)
4. Alexander Grothendieck, La théorie de Fredholm, Bulletin de la Société Mathématique de France 84 (1956), 319-384, doi:10.24033/bsmf.1476, https://numdam.org/item/BSMF_1956__84__319_0/ (the archimedean presentation with factorial denominators that Serre says cannot be followed in the p-adic setting)
5. Anthony F. Ruston, Fredholm Theory in Banach Spaces, Cambridge Tracts in Mathematics 86, Cambridge University Press, Cambridge, 1986, book doi:10.1017/cbo9780511569180, isbn 9780521248464, https://doi.org/10.1017/cbo9780511569180 (chapter 'Fredholm formulae and the Riesz theory', pp. 135-165, doi:10.1017/cbo9780511569180.007; the archimedean reference fixing what 'Cramer-type Fredholm formulae' means)
6. Bernard Dwork, On the zeta function of a hypersurface, Publications Mathématiques de l'IHÉS 12 (1962), 5-68, doi:10.1007/BF02684275, https://numdam.org/item/PMIHES_1962__12__5_0/ (the application target of Serre's Section 9)
7. Dietmar Treiber, Über orthogonalisierbare nicht-archimedische Banachräume, Archiv der Mathematik 24 (1973), 71-80, doi:10.1007/bf01228177, https://doi.org/10.1007/bf01228177 (later work on which non-archimedean Banach spaces admit orthonormal bases, the structure-theoretic line adjacent to Serre's c(I) restriction)
8. C. Perez-Garcia and S. Vega, Perturbation theory of p-adic Fredholm and semi-Fredholm operators, Indagationes Mathematicae 15 (2004), no. 1, 115-127, doi:10.1016/S0019-3577(04)90009-2, https://doi.org/10.1016/S0019-3577(04)90009-2 (representative of the post-1966 p-adic Fredholm operator literature)
9. M. M. Vishik, Nonarchimedean spectral theory, Journal of Soviet Mathematics 30 (1985), 2513-2555, doi:10.1007/bf02249122, https://doi.org/10.1007/bf02249122 (the non-archimedean spectral-theory line via the Shnirel'man integral)