Cauchy Problem of the Non-self-adjoint Gauss-laguerre Semigroups and Uniform Bounds of Generalized Laguerre Polynomials

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 on non-harmonic analysis. Our methodology also allows to provide smoothness properties for the semigroup as well as for the heat kernel. The biorthogonal sequences appearing in the eigenvalues expansions can be both expressed in terms of sequences of polynomials, generalizing the Laguerre polynomials. By means of a delicate saddle point method, we provide uniform asymptotic bounds allowing us to get an upper bound for their norms in the weighted Hilbert space. We believe that this work opens a way to construct spectral expansion for more general non-self adjoint Markov semigroups.