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We propose a new approach to construct an eigenvalue expansions in a weighted Hilbert space of the solution to the Cauchy problem associated to the so-called Gauss-Laguerre contraction semigroups, whose generators turns out to be a natural non-self-adjoint and nonlocal generalization of the Laguerre differential operators. Our methods rely on an intertwining relationship that we establish between this semigroup and the one of the classical Laguerre semigroup, combined with techniques based ondoi:10.4171/jst/178 fatcat:npppws2dsbbgbjqwpj56ecu33a