We show several examples of n.a. valued fields with involution. Then, by means of a field of this kind, we introduce “n.a. Hilbert spaces” in which the norm comes from a certain hermitian sesquilinear form. We study these spaces and the algebra of bounded operators which are defined on them and have an adjoint. Essential differences with respect to the usual case are observed.

Dual codes are defined with respect to non-degenerate sesquilinear or bilinear forms over a finite Frobenius ring. These dual codes have the properties one expects from a dual code: they satisfy a double-dual property, they have cardinality complementary to that of the primal code, and they satisfy the MacWilliams identities for the Hamming weight. 2010 MSC: 94B05, 15A63

We discuss techniques for analysing the structure of the group obtained by reducing the image of the Burau representation of the braid group modulo a prime. The main tools are a certain sesquilinear form first introduced by Squier and consideration of the action of the group on a Euclidean building. AMS Classification 20F36; 57M07 57M25

We provide a systematic study of sesquilinear hermitian forms and a new proof of the calculus of some exponential sums defined with quadratic her-mitian forms. The computation of the number of solutions of equations such as Tr Ft/Fs (f (x) + v.x) = 0 or Tr Ft/Fs (f (x)) = a allows us to construct codes and to obtain their parameters.

The aim of this paper is to study uniformly elliptic operators with general Wentzell boundary conditions in suitable Lp-spaces by using the approach of sesquilinear forms. We use different tools to re-prove analiticity and related results concerning the semigroups generated by the above operators. In addition, we make some complementary observations on, among other things, compactness issues an...

We discuss a Hilbert space method that allows to prove analytical well-posedness of a class of linear strongly damped wave equations. The main technical tool is a perturbation lemma for sesquilinear forms, which seems to be new. In most common linear cases we can furthermore apply a recent result due to Crouzeix–Haase, thus extending several known results and obtaining optimal analyticity angle.