Joint Hyponormality of Rational Toeplitz Pairs

Abstract

We characterize hyponormal “rational” Toeplitz pairs which are pairs of Toeplitz operators whose symbols are rational functions in L∞. The main result of this article is as follows. If T = (Tϕ, Tψ) is a hyponormal rational Toeplitz pair then ϕ − βψ ∈H2 for some constant β; in other words, their co-analytic parts necessarily coincide up to a constant multiple. As a corollary we get a complete characterization of hyponormal rational Toeplitz pairs.