A hyperbolic universal operator commuting with a compact operator

A Hilbert space operator is called universal (in the sense of Rota) if every on similar to a multiple restriction universal one its invariant subspaces. We exhibit an analytic Toeplitz whose adjoint in Rota and which commutes with quasinilpotent, injective, compact dense range, but unlike other examples, it acts Bergman instead Hardy this associated “hyperbolic” composition operator.