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In this paper, we introduce and study non-local Jacobi operators, whichgeneralize the classical (local) Jacobi operators. We show that these operatorsextend to generators of ergodic Markov semigroups with unique invariantprobability measures and study their spectral and convergence properties. Inparticular, we derive a series expansion of the semigroup in terms ofexplicitly defined polynomials, which generalize the classical Jacobiorthogonal polynomials. In addition, we give a complete characterization of thespectrum of the non-self-adjoint generator and semigroup. We show that thevariance decay of the semigroup is hypocoercive with explicit constants, whichprovides a natural generalization of the spectral gap estimate. After a randomwarm-up time, the semigroup also decays exponentially in entropy and is bothhypercontractive and ultracontractive. Our proofs hinge on the development ofcommutation identities, known as intertwining relations, between local andnon-local Jacobi operators and semigroups, with the local objects serving asreference points for transferring properties from the local to the non-localcase.

On non-local ergodic Jacobi semigroups: spectral theory, convergence-to-equilibrium and contractivity