Partial *-algebras of Distributions

Abstract

The problem of multiplying elements of the conjugate dual of certain kind of commutative generalized Hilbert algebras, which are dense in the set of C ∞ -vectors of a self-adjoint operator, is considered in the framework of the so-called duality method. The multiplication is defined by identifying each distribution with a multiplication operator acting on the natural rigged Hilbert space. Certain spaces, that are an abstract version of the Bessel potential spaces, are used to factorize the product. §1. Introduction Distributions are, as is well-known, typical objects that can be multiplied only if some very particular circumstances occur. Nevertheless, products of distributions, sometimes understood only in formal sense, frequently appear in physical applications (for instance in quantum field theory) and play a relevant role in the theory of partial differential equations. For these reasons many possibilities of defining a (partial) multiplication have been suggested in the literature (see [1] for an overview) dating back to the famous Schwartz paper devoted to the impossibility of multiplying two Dirac delta measures massed at the same point [2].Reconsidering an idea developed in [3], we study in this paper the possibility of making of the space S (R n ) a partial *-algebra in the sense of [4]. As is clear, the multiplication of a test function times a tempered distribution, makes of (S (R n ), S(R n )) a quasi*-algebra in the sense of Lassner [5,6] Given a rigged Hilbert space, D ⊂ H ⊂ D , we denote with L(D, D ) the set of all continuous linear maps fromD) (for this reason we denote with † both involutions).uous. Hence, the problem of multiplying two distributions U, V ∈ S (R n ) can be viewed in terms of multiplication of the corresponding operators, S (R n )) and then investigating conditions under which L U ·L V = L W for some W ∈ S (R n ). This was partially done in [3] also to a certain degree of abstractness: therein, in fact, tempered distributions were considered as a special case of the so called A-distributions defined as elements of the (conjugate)Partial *-algebras of Distributions

Operators in rigged Hilbert spaces: Some spectral properties

“…Spaces of linear maps acting on a rigged Hilbert space (RHS, for short) D ⊂ H ⊂ D × have often been considered in the literature both from a pure mathematical point of view [22,23,28,31] and for their applications to quantum theories (generalized eigenvalues, resonances of Schrödinger operators, quantum fields...) [8,13,12,15,14,16,25]. The spaces of test functions and the distributions over them constitute relevant examples of rigged Hilbert spaces and operators acting on them are a fundamental tool in several problems in Analysis (differential operators with singular coefficients, Fourier transforms) and also provide the basic background for the study of the problem of the multiplication of distributions by the duality method [24,27,34].…”