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The bispectral problem is motivated by an effort to understand and extend aremarkable phenomenon in Fourier analysis on the real line: the operator oftime-and-band limiting is an integral operator admitting a second-orderdifferential operator with a simple spectrum in its commutator. In thisarticle, we discuss a noncommutative version of the bispectral problem,obtained by allowing all objects in the original formulation to bematrix-valued. Deep attention is given to bispectral algebras and theirpresentations as a tool to get information about bispectral triples.

Matrix Bispectrality and Noncommutative Algebras: beyond the prolate spheroidals