The δ Equation on a Hilbert Space and Some Applications to Complex Analysis on Infinite Dimensional Vector Spaces

Abstract

This chapter at first mentions the importance of the problem in C N , and with respect to other work concerning Complex Analysis on infinite dimensional spaces. It is possible to solve the equation on a Hilbert space in a weak sense, one can find a regular solution only on a subspace of H (more precisely on the image of a Hilbert–Schmidt operator); this fact is essentially, because of the absence of a measure, which is invariant by the translations on an infinite dimensional Hilbert space. This phenomenon has similar consequences in other questions of infinite dimensional Analysis, such as potential theory, or parabolic equation. However, the result obtained can be useful in Complex Analysis on other types of infinite dimensional vector spaces; one can show for instance, that it is possible to solve Cousin's first problem on the dual of a Fréchet nuclear space. The chapter presents a discussion on weak resolution of the equation. Let H be a separable, complex, infinite dimensional Hilbert space. It is known that there is no measure on H, which is invariant by translation of analysis on infinite dimensional vector spaces; gaussian measure will take the part of Lebesgue measure on R N .