  The subject of time-band-limiting, originating in signal processing, is
dominated by the miracle that a naturally appearing integral operator admits a
commuting differential one allowing for a numerically efficient way to compute
its eigenfunctions. Bispectrality is an effort to dig into the reasons behind
this miracle and goes back to joint work with H. Duistermaat. This search has
revealed unexpected connections with several parts of mathematics, including
integrable systems.
  Here we consider a matrix valued version of bispectrality and give a general
condition under which we can display a constructive and simple way to obtain
the commuting differential operator.
  Furthermore, we build an operator that commutes with both the time-limiting
operator and the band-limiting operators.
