Rigged Hilbert spaces and contractive families of Hilbert spaces

Abstract

Abstract. The existence of a rigged Hilbert space whose extreme spaces are, respectively, the projective and the inductive limit of a directed contractive family of Hilbert spaces is investigated. It is proved that, when it exists, this rigged Hilbert space is the same as the canonical rigged Hilbert space associated to a family of closable operators in the central Hilbert space.

Singular Perturbations and Operators in Rigged Hilbert Spaces

“…As it has been shown in [5,6], we know that if X ∈ L B (D, D × ) then there exists A ∈ L † (D) such that…”

Hermite Functions and Fourier Series

“…The second identity in (71) makes use of the Lebesgue-dominated convergence theorem [32] in order to interchange the integral and the series as anticipated in (15). We have also used the change of variable s = φ + 2kπ and e imφ = e im(φ+2kπ) = e ims .…”

“…The formalism of rigged Hilbert spaces was introduced by Gelfand [7]. Although this is not quite familiar to many theoretical physicists, it has acquired more and more importance in the field of mathematical physics [8][9][10][11][12][13] and even in mathematics [14][15][16][17][18][19][20].…”

Hermite functions and Fourier series

“…The rigging of Hilbert spaces is necessary for the continuity of significant operators we are using here such as ladder operators. Although the formalism of rigged Hilbert spaces is not new as was introduced by Gelfand [7], it has acquired more and more importance by mathematical physicists [8][9][10][11][12][13] and more recently by mathematicians [14][15][16][17][18][19][20]. Furthermore, the authors have published a series of papers in which it is shown the close connection between Lie algebras, special functions, discrete and continuous basis and rigged Hilbert spaces [22][23][24][25][26].…”