Generalized Fock spaces and the Stirling numbers

TL;DR

This paper extends the Bargmann-Fock-Segal space into a Gelfand triple, characterizing test functions via Stirling numbers and exploring the algebraic structure of its dual space, with implications for mathematical physics.

Contribution

It introduces a new embedding of the Fock space into a Gelfand triple and characterizes the test function space using Stirling numbers of the second kind.

Findings

Characterization of test functions in the Gelfand triple

Topological algebra structure of the dual space

Geometric and algebraic descriptions of the space of test functions

Abstract

The Bargmann-Fock-Segal space plays an important role in mathematical physics, and has been extended into a number of directions. In the present paper we imbed this space into a Gelfand triple. The spaces forming the Fr\'echet part (i.e. the space of test functions) of the triple are characterized both in a geometric way and in terms of the adjoint of multiplication by the complex variable, using the Stirling numbers of the second kind. The dual of the space of test functions has a topological algebra structure, of the kind introduced and studied by the first named author and G. Salomon.