Closed Intertwining Operators on locally algebraic principal series

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Number Theory studies the solutions of polynomials over the field of rational numbers Q; p-adic Number Theory instead studies the solutions of polynomials over Q's topological completions:Besides the usual absolute value, there is on Q for each prime number p the p-adic absolute value that measures how often p divides an integer. Completing Q by this absolute value, we obtain the field of p-adic numbers Q_p. For a rational polynomial over Q, analytical arguments provide roots in Q_p outside of Q; the completion simplified Q_p arithmetically. Once the question of solvability is settled over every Q_p, so it is in many ways over Q (Hasse principle).Galois Theory studies this question by looking at the automorphisms of Q_p's field extensions. The absolute Galois group consists of all Q_p-algebra automorphisms of Q_p's algebraic closure. It encodes all arithmetic information on Q_p.We study it by looking at its actions on finite dimensional vector spaces. Here the p-adic Langlands program envisions a bridge between the mysterious arithmetical Galois group and the familiar linear group G=GL_n(Q_p); it formulates a correspondence between continuous Galois actions on n-dimensional vector spaces and uniformly continuous G-actions on Banach spaces.My research revolves around the construction of such a Banach space. There is a uniformly continuous G-action on B if and only if there is a G-invariant norm || on B. This is largely an open problem. The case GL_2(Q_p) was solved by Colmez and others by describing B, for a real number R>=0, through R-times differentiable (or C^R-)functions on the unit ball of Q_p.For more general cases my thesis introduced a C^R-Theory over p-adic manifolds, which has been examined in more detail. The newest preprint computes the Fourier coefficients of the C^R-functions on the unit ball in a finite extension F of Q_p. This serves towards a generalization of Colmez`s proof strategy to GL_2(F).If the G-action is locally given by rational functions, then the C^R-theory allows us to regard || as, a priori, seminorm on C^R-functions in many variables on G. This result is outlined in the notes of my lecture at the Séminaire Automorphe.Firstly I am revising the C^R-theory towards a handy description via Taylor polynomials, and compare it with other recent approaches to p-adic calculus.What is more, for || being a proper norm, special operators on C^R-function spaces, though surely not continuous, nevertheless must be closed. The notion of a closed operator is a familiar notion in classical Functional Analysis, but yet unheard-of in p-adic Functional Analysis. This problem I will investigate in special cases under assistance by Pierre Colmez and Ariane Mezard.