Subnormal Toeplitz operators and the kernels of their self-commutators

In this note we give a connection between subnormal Toeplitz operators and the kernels of their self-commutators. This is closely related to P.R. Halmosʹs Problem 5: Is every subnormal Toeplitz operator either normal or analytic? Our main theorem is as follows: If φ ∈ L ∞ is such that φ and φ ¯ are of bounded type (that is, they are quotients of two analytic functions on the open unit disk) and if the kernel of the self-commutator of T φ is invariant for T φ then T φ is either normal or analytic.