For , let be the Bergman space of holomorphic functions in the upper half-plane square-integrable with respect to the measure , where is the Lebesgue measure; the spaces can, in fact, be “analytically continued” to all . The Cayley transform induces an isomorphism , , between and the similar Bergman space on the unit disc with respect to the measure const.
The authors consider the restriction map on given by , . It turns out that is a densely defined, closed, and injective operator from into with dense range; hence, the partial isometry component in the polar decomposition is unitary. They identify explicitly, and show that , where are the Laguerre polynomials. This fact can then be used to obtain various recurrence formulas for .
The main idea underlying all these developments is the so-called “restriction principle” of B.Ørsted and the third author. For more information and other applications of this principle, see the papers of B. Ørsted and G. Zhang [Indiana Univ. Math. J. 43, 551-583 (1994; Zbl 0805.46053)] and G. Ólafsson and B. Ørsted [in: Lie theory and its applications in physics (Clausthal, 1995; Zbl 0916.22006)].