doi:10.1093/imrn/rnw186engGrünbaum, F.A.Velázquez, L.The CMV bispectral problemART-2017-104342A classical result due to Bochner classifies the orthogonal polynomials on the real line which are common eigenfunctions of a second order linear differential operator. We settle a natural version of the Bochner problem on the unit circle which answers a similar question concerning orthogonal Laurent polynomials and can be formulated as a bispectral problem involving CMV matrices. We solve this CMV bispectral problem in great generality proving that, except the Lebesgue measure, no other one on the unit circle yields a sequence of orthogonal Laurent polynomials which are eigenfunctions of a linear differential operator of arbitrary order. Actually, we prove that this is the case even if such an eigenfunction condition is imposed up to finitely many orthogonal Laurent polynomials.2017http://zaguan.unizar.es/record/6524810.1093/imrn/rnw186http://zaguan.unizar.es/record/65248oai:zaguan.unizar.es:65248info:eu-repo/grantAgreement/ES/DGA/E64info:eu-repo/grantAgreement/ES/MICINN/MTM2011-28952-C02-01info:eu-repo/grantAgreement/ES/MINECO/MTM2014-53963-PINTERNATIONAL MATHEMATICS RESEARCH NOTICES 2017, 19 (2017), 5833-5860by-nc-ndhttp://creativecommons.org/licenses/by-nc-nd/3.0/es/info:eu-repo/semantics/openAccess