A HYPERBOLIC UNIVERSAL OPERATOR COMMUTING WITH A COMPACT OPERATOR

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

Automated summary

In this article, we present an analytic Toeplitz operator and its adjoint, which are universal in the sense of Rota in the Hilbert space. They commute with a specific operator and act on the Bergman space instead of the Hardy space, being associated with a hyperbolic composition operator.