On the representation of linear operators in L2 spaces by means of ’’generalized matrices’’

Abstract

This paper concerns the representation of linear operators of L2 spaces by means of ’’generalized matrices’’ as it is usual, following Dirac, in quantum mechanics and in electronics. The known possibility of representing (on a nuclear test-function space) the bounded operators by means of distribution kernels is shown to extend to all the closable operators whose domain contains the test-function space (hence to all the Hermitian operators whose domain contains the Schwartz space 𝒟 of the infinitely differentiable functions with compact support). The representation of the adjoint operator is considered, and the possibility of representing the product of operators by means of a suitably defined ’’Volterra convolution’’ is studied. In particular it is shown that -algebras of unbounded operators (which are, for instance, generated by the canonical coordinates and momenta and the total energy of most quantum mechanical systems of n particles) may be represented isomorphically by means of distribution kernels, so that the Dirac’s rules on ’’generalized matrices’’ apply in the sense of distributions without further assumptions.