Resumen: A 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. Idioma: Inglés DOI: 10.1093/imrn/rnw186 Año: 2017 Publicado en: INTERNATIONAL MATHEMATICS RESEARCH NOTICES 2017, 19 (2017), 5833-5860 ISSN: 1073-7928 Factor impacto JCR: 1.145 (2017) Categ. JCR: MATHEMATICS rank: 52 / 309 = 0.168 (2017) - Q1 - T1 Factor impacto SCIMAGO: 2.168 - Mathematics (miscellaneous) (Q1)